2003
2003
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1.I.1B
Part IA, 2003 comment(a) Write the permutation
as a product of disjoint cycles. Determine its order. Compute its sign.
(b) Elements and of a group are conjugate if there exists a such that
Show that if permutations and are conjugate, then they have the same sign and the same order. Is the converse true? (Justify your answer with a proof or counterexample.)
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1.I.2D
Part IA, 2003 commentFind the characteristic equation, the eigenvectors , and the corresponding eigenvalues of the matrix
Show that spans the complex vector space .
Consider the four subspaces of defined parametrically by
Show that multiplication by maps three of these subspaces onto themselves, and the remaining subspace into a smaller subspace to be specified.
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1.II.5B
Part IA, 2003 comment(a) In the standard basis of , write down the matrix for a rotation through an angle about the origin.
(b) Let be a real matrix such that and , where is the transpose of .
(i) Suppose that has an eigenvector with eigenvalue 1 . Show that is a rotation through an angle about the line through the origin in the direction of , where trace .
(ii) Show that must have an eigenvector with eigenvalue 1 .
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1.II.6A
Part IA, 2003 commentLet be a linear map
Define the kernel and image of .
Let . Show that the equation has a solution if and only if
Let have the matrix
with respect to the standard basis, where and is a real variable. Find and for . Hence, or by evaluating the determinant, show that if and then the equation has a unique solution for all values of .
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1.II.7B
Part IA, 2003 comment(i) State the orbit-stabilizer theorem for a group acting on a set .
(ii) Denote the group of all symmetries of the cube by . Using the orbit-stabilizer theorem, show that has 48 elements.
Does have any non-trivial normal subgroups?
Let denote the line between two diagonally opposite vertices of the cube, and let
be the subgroup of symmetries that preserve the line. Show that is isomorphic to the direct product , where is the symmetric group on 3 letters and is the cyclic group of order 2 .
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1.II.8D
Part IA, 2003 commentLet and be non-zero vectors in . What is meant by saying that and are linearly independent? What is the dimension of the subspace of spanned by and if they are (1) linearly independent, (2) linearly dependent?
Define the scalar product for . Define the corresponding norm of . State and prove the Cauchy-Schwarz inequality, and deduce the triangle inequality.
By means of a sketch, give a geometric interpretation of the scalar product in the case , relating the value of to the angle between the directions of and .
What is a unit vector? Let be unit vectors in . Let
Show that
(i) for any fixed, linearly independent and , the minimum of over is attained when for some ;
(ii) in all cases;
(iii) and in the case where .
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3.I.1A
Part IA, 2003 commentThe mapping of into itself is a reflection in the plane . Find the matrix of with respect to any basis of your choice, which should be specified.
The mapping of into itself is a rotation about the line through , followed by a dilatation by a factor of 2 . Find the matrix of with respect to a choice of basis that should again be specified.
Show explicitly that
and explain why this must hold, irrespective of your choices of bases.
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3.I.2B
Part IA, 2003 commentShow that if a group contains a normal subgroup of order 3, and a normal subgroup of order 5 , then contains an element of order 15 .
Give an example of a group of order 10 with no element of order
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3.II.7B
Part IA, 2003 commentLet be the group of Möbius transformations of and let be a set of three distinct points in .
(i) Show that there exists a sending to to 1 , and to .
(ii) Hence show that if , then is isomorphic to , the symmetric group on 3 letters.
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3.II.8B
Part IA, 2003 comment(a) Determine the characteristic polynomial and the eigenvectors of the matrix
Is it diagonalizable?
(b) Show that an matrix with characteristic polynomial is diagonalizable if and only if .
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3.II.5E
Part IA, 2003 comment(a) Show, using vector methods, that the distances from the centroid of a tetrahedron to the centres of opposite pairs of edges are equal. If the three distances are and if are the distances from the centroid to the vertices, show that
[The centroid of points in with position vectors is the point with position vector
(b) Show that
with , is the equation of a right circular double cone whose vertex has position vector a, axis of symmetry and opening angle .
Two such double cones, with vertices and , have parallel axes and the same opening angle. Show that if , then the intersection of the cones lies in a plane with unit normal
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3.II.6E
Part IA, 2003 commentDerive an expression for the triple scalar product in terms of the determinant of the matrix whose rows are given by the components of the three vectors .
Use the geometrical interpretation of the cross product to show that , will be a not necessarily orthogonal basis for as long as .
The rows of another matrix are given by the components of three other vectors . By considering the matrix , where denotes the transpose, show that there is a unique choice of such that is also a basis and
Show that the new basis is given by
Show that if either or is an orthonormal basis then is a rotation matrix.
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1.I.3B
Part IA, 2003 commentDefine what it means for a function of a real variable to be differentiable at .
Prove that if a function is differentiable at , then it is continuous there.
Show directly from the definition that the function
is differentiable at 0 with derivative 0 .
Show that the derivative is not continuous at 0 .
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1.I.4C
Part IA, 2003 commentExplain what is meant by the radius of convergence of a power series.
Find the radius of convergence of each of the following power series: (i) , (ii) .
In each case, determine whether the series converges on the circle .
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1.II.9F
Part IA, 2003 commentProve the Axiom of Archimedes.
Let be a real number in , and let be positive integers. Show that the limit
exists, and that its value depends on whether is rational or irrational.
[You may assume standard properties of the cosine function provided they are clearly stated.]
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1.II.10F
Part IA, 2003 commentState without proof the Integral Comparison Test for the convergence of a series of non-negative terms.
Determine for which positive real numbers the series converges.
In each of the following cases determine whether the series is convergent or divergent: (i) , (ii) , (iii) .
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1.II.11B
Part IA, 2003 commentLet be continuous. Define the integral . (You are not asked to prove existence.)
Suppose that are real numbers such that for all . Stating clearly any properties of the integral that you require, show that
The function is continuous and non-negative. Show that
Now let be continuous on . By suitable choice of show that
and by making an appropriate change of variable, or otherwise, show that
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1.II.12C
Part IA, 2003 commentState carefully the formula for integration by parts for functions of a real variable.
Let be infinitely differentiable. Prove that for all and all ,
By considering the function at , or otherwise, prove that the series
converges to .
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2.I.1D
Part IA, 2003 commentConsider the equation
Using small line segments, sketch the flow directions in implied by the right-hand side of . Find the general solution (i) in , (ii) in .
Sketch a solution curve in each of the three regions , and .
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2.I.2D
Part IA, 2003 commentConsider the differential equation
where is a positive constant. By using the approximate finite-difference formula
where is a positive constant, and where denotes the function evaluated at for integer , convert the differential equation to a difference equation for .
Solve both the differential equation and the difference equation for general initial conditions. Identify those solutions of the difference equation that agree with solutions of the differential equation over a finite interval in the limit , and demonstrate the agreement. Demonstrate that the remaining solutions of the difference equation cannot agree with the solution of the differential equation in the same limit.
[You may use the fact that, for bounded .]
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2.II.5D
Part IA, 2003 comment(a) Show that if is an integrating factor for an equation of the form
then .
Consider the equation
in the domain . Using small line segments, sketch the flow directions in that domain. Show that is an integrating factor for the equation. Find the general solution of the equation, and sketch the family of solutions that occupies the larger domain .
(b) The following example illustrates that the concept of integrating factor extends to higher-order equations. Multiply the equation
by , and show that the result takes the form , for some function to be determined. Find a particular solution such that with finite at , and sketch its graph in .
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Part IA, 2003
commentDefine the Wronskian associated with solutions of the equation
and show that
Evaluate the expression on the right when .
Given that and that , show that solutions in the form of power series,
can be found if and only if or 3 . By constructing and solving the appropriate recurrence relations, find the coefficients for each power series.
You may assume that the equation is satisfied by and by . Verify that these two solutions agree with the two power series found previously, and that they give the found previously, up to multiplicative constants.
[Hint:
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2.II.7D
Part IA, 2003 commentConsider the linear system
where the -vector and the matrix are given; has constant real entries, and has distinct eigenvalues and linearly independent eigenvectors . Find the complementary function. Given a particular integral , write down the general solution. In the case show that the complementary function is purely oscillatory, with no growth or decay, if and only if
Consider the same case with trace and and with
where are given real constants. Find a particular integral when
(i) and ;
(ii) but .
In the case
with , find the solution subject to the initial condition at .
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2.II.8D
Part IA, 2003 commentFor all solutions of
show that where
In the case , analyse the properties of the critical points and sketch the phase portrait, including the special contours for which . Comment on the asymptotic behaviour, as , of solution trajectories that pass near each critical point, indicating whether or not any such solution trajectories approach from, or recede to, infinity.
Briefly discuss how the picture changes when is made small and positive, using your result for to describe, in qualitative terms, how solution trajectories cross -contours.
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4.I.3E
Part IA, 2003 commentBecause of an accident on launching, a rocket of unladen mass lies horizontally on the ground. It initially contains fuel of mass , which ignites and is emitted horizontally at a constant rate and at uniform speed relative to the rocket. The rocket is initially at rest. If the coefficient of friction between the rocket and the ground is , and the fuel is completely burnt in a total time , show that the final speed of the rocket is
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4.I.4E
Part IA, 2003 commentWrite down an expression for the total momentum and angular momentum with respect to an origin of a system of point particles of masses , position vectors (with respect to , and velocities .
Show that with respect to a new origin the total momentum and total angular momentum are given by
and hence
where is the constant vector displacement of with respect to . How does change under change of origin?
Hence show that either
(1) the total momentum vanishes and the total angular momentum is independent of origin, or
(2) by choosing in a way that should be specified, the total angular momentum with respect to can be made parallel to the total momentum.
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4.II.9E
Part IA, 2003 commentWrite down the equation of motion for a point particle with mass , charge , and position vector moving in a time-dependent magnetic field with vanishing electric field, and show that the kinetic energy of the particle is constant. If the magnetic field is constant in direction, show that the component of velocity in the direction of is constant. Show that, in general, the angular momentum of the particle is not conserved.
Suppose that the magnetic field is independent of time and space and takes the form and that is the rate of change of area swept out by a radius vector joining the origin to the projection of the particle's path on the plane. Obtain the equation
where are plane polar coordinates. Hence obtain an equation replacing the equation of conservation of angular momentum.
Show further, using energy conservation and , that the equations of motion in plane polar coordinates may be reduced to the first order non-linear system
where and are constants.
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4.II.10E
Part IA, 2003 commentWrite down the equations of motion for a system of gravitating particles with masses , and position vectors .
The particles undergo a motion for which , where the vectors are independent of time . Show that the equations of motion will be satisfied as long as the function satisfies
where is a constant and the vectors satisfy
Show that has as first integral
where is another constant. Show that
where is the gradient operator with respect to and
Using Euler's theorem for homogeneous functions (see below), or otherwise, deduce that
Hence show that all solutions of satisfy
where
Deduce that must be positive and that the total kinetic energy plus potential energy of the system of particles is equal to .
[Euler's theorem states that if
then
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4.II.11E
Part IA, 2003 commentState the parallel axis theorem and use it to calculate the moment of inertia of a uniform hemisphere of mass and radius about an axis through its centre of mass and parallel to the base.
[You may assume that the centre of mass is located at a distance a from the flat face of the hemisphere, and that the moment of inertia of a full sphere about its centre is , with .]
The hemisphere initially rests on a rough horizontal plane with its base vertical. It is then released from rest and subsequently rolls on the plane without slipping. Let be the angle that the base makes with the horizontal at time . Express the instantaneous speed of the centre of mass in terms of and the rate of change of , where is the instantaneous distance from the centre of mass to the point of contact with the plane. Hence write down expressions for the kinetic energy and potential energy of the hemisphere and deduce that
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4.II.12E
Part IA, 2003 commentLet be plane polar coordinates and and unit vectors in the direction of increasing and respectively. Show that the velocity of a particle moving in the plane with polar coordinates is given by
and that the unit normal to the particle path is parallel to
Deduce that the perpendicular distance from the origin to the tangent of the curve is given by
The particle, whose mass is , moves under the influence of a central force with potential . Use the conservation of energy and angular momentum to obtain the equation
Hence express as a function of as the integral
where
Evaluate the integral and describe the orbit when , with a positive constant.
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4.I.1C
Part IA, 2003 comment(i) Prove by induction or otherwise that for every ,
(ii) Show that the sum of the first positive cubes is divisible by 4 if and only if or .
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4.I.2C
Part IA, 2003 commentWhat is an equivalence relation? For each of the following pairs , determine whether or not is an equivalence relation on :
(i) iff is an even integer;
(ii) iff ;
(iii) iff ;
(iv) iff is times a perfect square.
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4.II.5C
Part IA, 2003 commentDefine what is meant by the term countable. Show directly from your definition that if is countable, then so is any subset of .
Show that is countable. Hence or otherwise, show that a countable union of countable sets is countable. Show also that for any is countable.
A function is periodic if there exists a positive integer such that, for every . Show that the set of periodic functions is countable.
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4.II.6C
Part IA, 2003 comment(i) Prove Wilson's theorem: if is prime then .
Deduce that if then
(ii) Suppose that is a prime of the form . Show that if then .
(iii) Deduce that if is an odd prime, then the congruence
has exactly two solutions ( if , and none otherwise.
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4.II.7C
Part IA, 2003 commentLet be integers. Explain what is their greatest common divisor . Show from your definition that, for any integer .
State Bezout's theorem, and use it to show that if is prime and divides , then divides at least one of and .
The Fibonacci sequence is defined by and for . Prove:
(i) and for every ;
(ii) and for every ;
(iii) if then .
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4.II.8C
Part IA, 2003 commentLet be a finite set with elements. How many functions are there from to ? How many relations are there on ?
Show that the number of relations on such that, for each , there exists at least one with , is .
Using the inclusion-exclusion principle or otherwise, deduce that the number of such relations for which, in addition, for each , there exists at least one with , is
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2.I.3F
Part IA, 2003 comment(a) Define the probability generating function of a random variable. Calculate the probability generating function of a binomial random variable with parameters and , and use it to find the mean and variance of the random variable.
(b) is a binomial random variable with parameters and is a binomial random variable with parameters and , and and are independent. Find the distribution of ; that is, determine for all possible values of .
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2.I.4F
Part IA, 2003 commentThe random variable is uniformly distributed on the interval . Find the distribution function and the probability density function of , where
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2.II.9F
Part IA, 2003 commentState the inclusion-exclusion formula for the probability that at least one of the events occurs.
After a party the guests take coats randomly from a pile of their coats. Calculate the probability that no-one goes home with the correct coat.
Let be the probability that exactly guests go home with the correct coats. By relating to , or otherwise, determine and deduce that
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2.II.10F
Part IA, 2003 commentThe random variables and each take values in , and their joint distribution is given by
Find necessary and sufficient conditions for and to be (i) uncorrelated; (ii) independent.
Are the conditions established in (i) and (ii) equivalent?
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2.II.11F
Part IA, 2003 commentA laboratory keeps a population of aphids. The probability of an aphid passing a day uneventfully is . Given that a day is not uneventful, there is probability that the aphid will have one offspring, probability that it will have two offspring and probability that it will die, where . Offspring are ready to reproduce the next day. The fates of different aphids are independent, as are the events of different days. The laboratory starts out with one aphid.
Let be the number of aphids at the end of the first day. What is the expected value of ? Determine an expression for the probability generating function of .
Show that the probability of extinction does not depend on , and that if then the aphids will certainly die out. Find the probability of extinction if and .
[Standard results on branching processes may be used without proof, provided that they are clearly stated.]
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2.II.12F
Part IA, 2003 commentPlanet Zog is a ball with centre . Three spaceships and land at random on its surface, their positions being independent and each uniformly distributed on its surface. Calculate the probability density function of the angle formed by the lines and .
Spaceships and can communicate directly by radio if , and similarly for spaceships and and spaceships and . Given angle , calculate the probability that can communicate directly with either or . Given angle , calculate the probability that can communicate directly with both and . Hence, or otherwise, show that the probability that all three spaceships can keep in in touch (with, for example, communicating with via if necessary) is .
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3.I.3A
Part IA, 2003 commentSketch the curve . By finding a parametric representation, or otherwise, determine the points on the curve where the radius of curvature is least, and compute its value there.
[Hint: you may use the fact that the radius of curvature of a parametrized curve is .]
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3.I.4A
Part IA, 2003 commentSuppose is a region in , bounded by a piecewise smooth closed surface , and is a scalar field satisfying
Prove that is determined uniquely in .
How does the situation change if the normal derivative of rather than itself is specified on ?
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3.II.9A
Part IA, 2003 commentLet be the closed curve that is the boundary of the triangle with vertices at the points and .
Specify a direction along and consider the integral
where . Explain why the contribution to the integral is the same from each edge of , and evaluate the integral.
State Stokes's theorem and use it to evaluate the surface integral
the components of the normal to being positive.
Show that in the above surface integral can be written in the form .
Use this to verify your result by a direct calculation of the surface integral.
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3.II.10A
Part IA, 2003 commentWrite down an expression for the Jacobian of a transformation
Use it to show that
where is mapped one-to-one onto , and
Find a transformation that maps the ellipsoid ,
onto a sphere. Hence evaluate
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3.II.11A
Part IA, 2003 comment(a) Prove the identity
(b) If is an irrotational vector field everywhere , prove that there exists a scalar potential such that .
Show that
is irrotational, and determine the corresponding potential .
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3.II.12A
Part IA, 2003 commentState the divergence theorem. By applying this to , where is a scalar field in a closed region in bounded by a piecewise smooth surface , and an arbitrary constant vector, show that
A vector field satisfies
By applying the divergence theorem to , prove Gauss's law
where is the piecewise smooth surface bounding the volume .
Consider the spherically symmetric solution
where . By using Gauss's law with a sphere of radius , centre , in the two cases and , show that
The scalar field satisfies . Assuming that as , and that is continuous at , find everywhere.
By using a symmetry argument, explain why is clearly satisfied for this if is any sphere centred at the origin.
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1.I.1F
Part IB, 2003 commentLet be a subset of . Prove that the following conditions on are equivalent:
(i) is closed and bounded.
(ii) has the Bolzano-Weierstrass property (i.e., every sequence in has a subsequence convergent to a point of ).
(iii) Every continuous real-valued function on is bounded.
[The Bolzano-Weierstrass property for bounded closed intervals in may be assumed.]
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1.II.10F
Part IB, 2003 commentExplain briefly what is meant by a metric space, and by a Cauchy sequence in a metric space.
A function is called a pseudometric on if it satisfies all the conditions for a metric except the requirement that implies . If is a pseudometric on , show that the binary relation on defined by is an equivalence relation, and that the function induces a metric on the set of equivalence classes.
Now let be a metric space. If and are Cauchy sequences in , show that the sequence whose th term is is a Cauchy sequence of real numbers. Deduce that the function defined by
is a pseudometric on the set of all Cauchy sequences in . Show also that there is an isometric embedding (that is, a distance-preserving mapping) , where is the equivalence relation on induced by the pseudometric as in the previous paragraph. Under what conditions on is bijective? Justify your answer.
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2.I.1F
Part IB, 2003 commentExplain what it means for a function to be differentiable at a point . Show that if the partial derivatives and exist in a neighbourhood of and are continuous at then is differentiable at .
Let
and . Do the partial derivatives of exist at Is differentiable at Justify your answers.
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2.II.10F
Part IB, 2003 commentLet be the space of real matrices. Show that the function
(where denotes the usual Euclidean norm on ) defines a norm on . Show also that this norm satisfies for all and , and that if then all entries of have absolute value less than . Deduce that any function such that is a polynomial in the entries of is continuously differentiable.
Now let be the mapping sending a matrix to its determinant. By considering as a polynomial in the entries of , show that the derivative is the function . Deduce that, for any is the mapping , where is the adjugate of , i.e. the matrix of its cofactors.
[Hint: consider first the case when is invertible. You may assume the results that the set of invertible matrices is open in and that its closure is the whole of , and the identity .]
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3.I.1F
Part IB, 2003 commentLet be the vector space of continuous real-valued functions on . Show that the function
defines a norm on .
Let . Show that is a Cauchy sequence in . Is convergent? Justify your answer.
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Part IB, 2003
commentState and prove the Contraction Mapping Theorem.
Let be a bounded metric space, and let denote the set of all continuous maps . Let be the function
Show that is a metric on , and that is complete if is complete. [You may assume that a uniform limit of continuous functions is continuous.]
Now suppose that is complete. Let be the set of contraction mappings, and let be the function which sends a contraction mapping to its unique fixed point. Show that is continuous. [Hint: fix and consider , where is close to .]
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4.I.1F
Part IB, 2003 commentExplain what it means for a sequence of functions to converge uniformly to a function on an interval. If is a sequence of continuous functions converging uniformly to on a finite interval , show that
Let . Does uniformly on Does ? Justify your answers.
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4.II.10F
Part IB, 2003 commentLet be a sequence of continuous complex-valued functions defined on a set , and converging uniformly on to a function . Prove that is continuous on .
State the Weierstrass -test for uniform convergence of a series of complex-valued functions on a set .
Now let , where
Prove carefully that is continuous on .
[You may assume the inequality
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1.I.7B
Part IB, 2003 commentLet and be a pair of conjugate harmonic functions in a domain .
Prove that
also form a pair of conjugate harmonic functions in .
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2.I.7B
Part IB, 2003 comment(a) Using the residue theorem, evaluate
(b) Deduce that
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2.II.16B
Part IB, 2003 comment(a) Show that if satisfies the equation
where is a constant, then its Fourier transform satisfies the same equation, i.e.
(b) Prove that, for each , there is a polynomial of degree , unique up to multiplication by a constant, such that
is a solution of for some .
(c) Using the fact that satisfies for some constant , show that the Fourier transform of has the form
where is also a polynomial of degree .
(d) Deduce that the are eigenfunctions of the Fourier transform operator, i.e. for some constants
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1.II.16B
Part IB, 2003 commentSketch the region which is the intersection of the discs
Find a conformal mapping that maps onto the right half-plane . Also find a conformal mapping that maps onto .
[Hint: You may find it useful to consider maps of the form .]
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4.I.8B
Part IB, 2003 commentFind the Laurent series centred on 0 for the function
in each of the domains (a) , (b) , (c) .
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4.II.17B
Part IB, 2003 commentLet
and let be the boundary of the domain
(a) Using the residue theorem, determine
(b) Show that the integral of along the circular part of tends to 0 as .
(c) Deduce that
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1.I.6C
Part IB, 2003 commentAn unsteady fluid flow has velocity field given in Cartesian coordinates by , where denotes time. Dye is released into the fluid from the origin continuously. Find the position at time of the dye particle that was released at time and hence show that the dye streak lies along the curve
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1.II.15C
Part IB, 2003 commentStarting from the Euler equations for incompressible, inviscid flow
derive the vorticity equation governing the evolution of the vorticity .
Consider the flow
in Cartesian coordinates , where is time and is a constant. Compute the vorticity and show that it evolves in time according to
where is the initial magnitude of the vorticity and is a unit vector in the -direction.
Show that the material curve that takes the form
at is given later by
where the function is to be determined.
Calculate the circulation of around and state how this illustrates Kelvin's circulation theorem.
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3.I.8C
Part IB, 2003 commentShow that the velocity field
where and in Cartesian coordinates , represents the combination of a uniform flow and the flow due to a line vortex. Define and evaluate the circulation of the vortex.
Show that
where is a circle const. Explain how this result is related to the lift force on a two-dimensional aerofoil or other obstacle.
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3.II.18C
Part IB, 2003 commentState the form of Bernoulli's theorem appropriate for an unsteady irrotational motion of an inviscid incompressible fluid in the absence of gravity.
Water of density is driven through a tube of length and internal radius by the pressure exerted by a spherical, water-filled balloon of radius attached to one end of the tube. The balloon maintains the pressure of the water entering the tube at in excess of atmospheric pressure, where is a constant. It may be assumed that the water exits the tube at atmospheric pressure. Show that
Solve equation ( ), by multiplying through by or otherwise, to obtain
where and is the initial radius of the balloon. Hence find the time when .
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4.I.7C
Part IB, 2003 commentInviscid fluid issues vertically downwards at speed from a circular tube of radius a. The fluid falls onto a horizontal plate a distance below the end of the tube, where it spreads out axisymmetrically.
Show that while the fluid is falling freely it has speed
and occupies a circular jet of radius
where is the height above the plate and is the acceleration due to gravity.
Show further that along the plate, at radial distances (i.e. far from the falling jet), where the fluid is flowing almost horizontally, it does so as a film of height , where
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4.II.16C
Part IB, 2003 commentDefine the terms irrotational flow and incompressible flow. The two-dimensional flow of an incompressible fluid is given in terms of a streamfunction as
in Cartesian coordinates . Show that the line integral
along any path joining the points and , where is the unit normal to the path. Describe how this result is related to the concept of mass conservation.
Inviscid, incompressible fluid is contained in the semi-infinite channel , , which has rigid walls at and at , apart from a small opening at the origin through which the fluid is withdrawn with volume flux per unit distance in the third dimension. Show that the streamfunction for irrotational flow in the channel can be chosen (up to an additive constant) to satisfy the equation
and boundary conditions
if it is assumed that the flow at infinity is uniform. Solve the boundary-value problem above using separation of variables to obtain
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Part IB, 2003
commentLet be the collection of all subsets such that or is finite. Let be the collection of all subsets of of the form , together with the empty set. Prove that and are both topologies on .
Show that a function from the topological space to the topological space is continuous if and only if one of the following alternatives holds:
(i) as ;
(ii) there exists such that for all but finitely many and for all .
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2.II.13E
Part IB, 2003 comment(a) Let be defined by and let be the image of . Prove that is compact and path-connected. [Hint: you may find it helpful to set
(b) Let be defined by , let be the image of and let be the closed unit . Prove that is connected. Explain briefly why it is not path-connected.
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Part IB, 2003
comment(a) Let be an analytic function such that for every . Prove that is constant.
(b) Let be an analytic function such that for every . Prove that is constant.
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3.II.13E
Part IB, 2003 comment(a) State Taylor's Theorem.
(b) Let and be defined whenever . Suppose that as , that no equals and that for every . Prove that for every .
(c) Let be a domain, let and let be a sequence of points in that converges to , but such that no equals . Let and be analytic functions such that for every . Prove that for every .
(d) Let be the domain . Give an example of an analytic function such that for every positive integer but is not identically 0 .
(e) Show that any function with the property described in (d) must have an essential singularity at the origin.
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4.I.4E
Part IB, 2003 comment(a) State and prove Morera's Theorem.
(b) Let be a domain and for each let be an analytic function. Suppose that is another function and that uniformly on . Prove that is analytic.
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4.II.13E
Part IB, 2003 comment(a) State the residue theorem and use it to deduce the principle of the argument, in a form that involves winding numbers.
(b) Let . Find all such that and . Calculate for each such . [It will be helpful to set . You may use the addition formulae and .]
(c) Let be the closed path . Use your answer to (b) to give a rough sketch of the path , paying particular attention to where it crosses the real axis.
(d) Hence, or otherwise, determine for every real the number of (counted with multiplicity) such that and . (You need not give rigorous justifications for your calculations.)
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1.I.4F
Part IB, 2003 commentDescribe the geodesics (that is, hyperbolic straight lines) in either the disc model or the half-plane model of the hyperbolic plane. Explain what is meant by the statements that two hyperbolic lines are parallel, and that they are ultraparallel.
Show that two hyperbolic lines and have a unique common perpendicular if and only if they are ultraparallel.
[You may assume standard results about the group of isometries of whichever model of the hyperbolic plane you use.]
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1.II.13F
Part IB, 2003 commentWrite down the Riemannian metric in the half-plane model of the hyperbolic plane. Show that Möbius transformations mapping the upper half-plane to itself are isometries of this model.
Calculate the hyperbolic distance from to , where and are positive real numbers. Assuming that the hyperbolic circle with centre and radius is a Euclidean circle, find its Euclidean centre and radius.
Suppose that and are positive real numbers for which the points and of the upper half-plane are such that the hyperbolic distance between them coincides with the Euclidean distance. Obtain an expression for as a function of . Hence show that, for any with , there is a unique positive value of such that the hyperbolic distance between and coincides with the Euclidean distance.
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3.I.4F
Part IB, 2003 commentShow that any isometry of Euclidean 3 -space which fixes the origin can be written as a composite of at most three reflections in planes through the origin, and give (with justification) an example of an isometry for which three reflections are necessary.
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3.II.14F
Part IB, 2003 commentState and prove the Gauss-Bonnet formula for the area of a spherical triangle. Deduce a formula for the area of a spherical -gon with angles . For what range of values of does there exist a (convex) regular spherical -gon with angle ?
Let be a spherical triangle with angles and where are integers, and let be the group of isometries of the sphere generated by reflections in the three sides of . List the possible values of , and in each case calculate the order of the corresponding group . If , show how to construct a regular dodecahedron whose group of symmetries is .
[You may assume that the images of under the elements of form a tessellation of the sphere.]
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1.I
Part IB, 2003 commentLet be the subset of consisting of all quintuples such that
and
Prove that is a subspace of . Solve the above equations for and in terms of and . Hence, exhibit a basis for , explaining carefully why the vectors you give form a basis.
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1.II.14E
Part IB, 2003 comment(a) Let be subspaces of a finite-dimensional vector space . Prove that
(b) Let and be finite-dimensional vector spaces and let and be linear maps from to . Prove that
(c) Deduce from this result that
(d) Let and suppose that . Exhibit linear maps such that and . Suppose that . Exhibit linear maps such that and .
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2.I.6E
Part IB, 2003 commentLet be distinct real numbers. For each let be the vector . Let be the matrix with rows and let be a column vector of size . Prove that if and only if . Deduce that the vectors .
[You may use general facts about matrices if you state them clearly.]
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2.II.15E
Part IB, 2003 comment(a) Let be an matrix and for each let be the matrix formed by the first columns of . Suppose that . Explain why the nullity of is non-zero. Prove that if is minimal such that has non-zero nullity, then the nullity of is 1 .
(b) Suppose that no column of consists entirely of zeros. Deduce from (a) that there exist scalars (where is defined as in (a)) such that for every , but whenever are distinct real numbers there is some such that .
(c) Now let and be bases for the same real dimensional vector space. Let be distinct real numbers such that for every the vectors are linearly dependent. For each , let be scalars, not all zero, such that . By applying the result of (b) to the matrix , deduce that .
(d) It follows that the vectors are linearly dependent for at most values of . Explain briefly how this result can also be proved using determinants.
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3.I.7G
Part IB, 2003 commentLet be an endomorphism of a finite-dimensional real vector space and let be another endomorphism of that commutes with . If is an eigenvalue of , show that maps the kernel of into itself, where is the identity map. Suppose now that is diagonalizable with distinct real eigenvalues where . Prove that if there exists an endomorphism of such that , then for all eigenvalues of .
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3.II.17G
Part IB, 2003 commentDefine the determinant of an complex matrix A. Let be the columns of , let be a permutation of and let be the matrix whose columns are . Prove from your definition of determinant that , where is the sign of the permutation . Prove also that
Define the adjugate matrix and prove from your that , where is the identity matrix. Hence or otherwise, prove that if , then is invertible.
Let and be real matrices such that the complex matrix is invertible. By considering as a function of or otherwise, prove that there exists a real number such that is invertible. [You may assume that if a matrix is invertible, then .]
Deduce that if two real matrices and are such that there exists an invertible complex matrix with , then there exists an invertible real matrix such that .
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4.I.6G
Part IB, 2003 commentLet be an endomorphism of a finite-dimensional real vector space such that . Show that can be written as the direct sum of the kernel of and the image of . Hence or otherwise, find the characteristic polynomial of in terms of the dimension of and the rank of . Is diagonalizable? Justify your answer.
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4.II.15G
Part IB, 2003 commentLet be a linear map between finite-dimensional vector spaces. Let
(a) Prove that and are subspaces of of dimensions
[You may use the result that there exist bases in and so that is represented by
where is the identity matrix and is the rank of
(b) Let be given by , where is the dual map induced by . Prove that is an isomorphism. [You may assume that is linear, and you may use the result that a finite-dimensional vector space and its dual have the same dimension.]
(c) Prove that
[You may use the results that and that can be identified with under the canonical isomorphism between a vector space and its double dual.]
(d) Conclude that .
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1.I.2D
Part IB, 2003 commentFermat's principle of optics states that the path of a light ray connecting two points will be such that the travel time is a minimum. If the speed of light varies continuously in a medium and is a function of the distance from the boundary , show that the path of a light ray is given by the solution to
where , etc. Show that the path of a light ray in a medium where the speed of light is a constant is a straight line. Also find the path from to if , and sketch it.
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1.II.11D
Part IB, 2003 comment(a) Determine the Green's function for the operator on with Dirichlet boundary conditions by solving the boundary value problem
when is not an integer.
(b) Use the method of Green's functions to solve the boundary value problem
when is not an integer.
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2.I.2C
Part IB, 2003 commentExplain briefly why the second-rank tensor
is isotropic, where is the surface of the unit sphere centred on the origin.
A second-rank tensor is defined by
where is the surface of the unit sphere centred on the origin. Calculate in the form
where and are to be determined.
By considering the action of on and on vectors perpendicular to , determine the eigenvalues and associated eigenvectors of .
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2.II.11C
Part IB, 2003 commentState the transformation law for an th-rank tensor .
Show that the fourth-rank tensor
is isotropic for arbitrary scalars and .
The stress and strain in a linear elastic medium are related by
Given that is symmetric and that the medium is isotropic, show that the stress-strain relationship can be written in the form
Show that can be written in the form , where is a traceless tensor and is a scalar to be determined. Show also that necessary and sufficient conditions for the stored elastic energy density to be non-negative for any deformation of the solid are that
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3.I.2D
Part IB, 2003 commentConsider the path between two arbitrary points on a cone of interior angle . Show that the arc-length of the path is given by
where . By minimizing the total arc-length between the points, determine the equation for the shortest path connecting them.
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3.II.12D
Part IB, 2003 commentThe transverse displacement of a stretched string clamped at its ends satisfies the equation
where is the wave velocity, and is the damping coefficient. The initial conditions correspond to a sharp blow at at time .
(a) Show that the subsequent motion of the string is given by
where .
(b) Describe what happens in the limits of small and large damping. What critical parameter separates the two cases?
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4.I.2D
Part IB, 2003 commentConsider the wave equation in a spherically symmetric coordinate system
where is the spherically symmetric Laplacian operator.
(a) Show that the general solution to the equation above is
where are arbitrary functions.
(b) Using separation of variables, determine the wave field in response to a pulsating source at the origin .
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4.II.11D
Part IB, 2003 commentThe velocity potential for inviscid flow in two dimensions satisfies the Laplace equation
(a) Using separation of variables, derive the general solution to the equation above that is single-valued and finite in each of the domains (i) ; (ii) .
(b) Assuming is single-valued, solve the Laplace equation subject to the boundary conditions at , and as . Sketch the lines of constant potential.
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2.I.5B
Part IB, 2003 commentLet
Find the LU factorization of the matrix and use it to solve the system .
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2.II.14B
Part IB, 2003 commentLet
be an approximation of the second derivative which is exact for , the set of polynomials of degree , and let
be its error.
(a) Determine the coefficients .
(b) Using the Peano kernel theorem prove that, for , the set of threetimes continuously differentiable functions, the error satisfies the inequality
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3.I.6B
Part IB, 2003 commentGiven distinct points , let
be the fundamental Lagrange polynomials of degree , let
and let be any polynomial of degree .
(a) Prove that .
(b) Hence or otherwise derive the formula
which is the decomposition of into partial fractions.
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3.II.16B
Part IB, 2003 commentThe functions are generated by the Rodrigues formula:
(a) Show that is a polynomial of degree , and that the are orthogonal with respect to the scalar product
(b) By induction or otherwise, prove that the satisfy the three-term recurrence relation
[Hint: you may need to prove the equality as well.]
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Part IB, 2003
commentTwo players A and B play a zero-sum game with the pay-off matrix
\begin{tabular}{r|rrr} & & & \ \hline & 4 & & \ & & 4 & 3 \ & & 6 & 2 \ & 3 & & \end{tabular}
Here, the entry of the matrix indicates the pay-off to player A if he chooses move and player chooses move . Show that the game can be reduced to a zero-sum game with pay-off matrix.
Determine the value of the game and the optimal strategy for player A.
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3.II.15H
Part IB, 2003 commentExplain what is meant by a transportation problem where the total demand equals the total supply. Write the Lagrangian and describe an algorithm for solving such a problem. Starting from the north-west initial assignment, solve the problem with three sources and three destinations described by the table
\begin{tabular}{|rrr|r|} \hline 5 & 9 & 1 & 36 \ 3 & 10 & 6 & 84 \ 7 & 2 & 5 & 40 \ \hline 14 & 68 & 78 & \ \hline \end{tabular}
where the figures in the box denote the transportation costs (per unit), the right-hand column denotes supplies, and the bottom row demands.
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4.I.5H
Part IB, 2003 commentState and prove the Lagrangian sufficiency theorem for a general optimization problem with constraints.
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4.II.14H
Part IB, 2003 commentUse the two-phase simplex method to solve the problem
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1.I.8G
Part IB, 2003 commentLet and be finite-dimensional vector spaces. Suppose that and are bilinear forms on and that is non-degenerate. Show that there exist linear endomorphisms of and of such that for all .
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1.II.17G
Part IB, 2003 comment(a) Suppose is an odd prime and an integer coprime to . Define the Legendre symbol and state Euler's criterion.
(b) Compute and prove that
whenever and are coprime to .
(c) Let be any integer such that . Let be the unique integer such that and . Prove that
(d) Find
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2.I.8G
Part IB, 2003 commentLet be a finite-dimensional real vector space and a positive definite symmetric bilinear form on . Let be a linear map such that for all and in . Prove that if is invertible, then the dimension of must be even. By considering the restriction of to its image or otherwise, prove that the rank of is always even.
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2.II.17G
Part IB, 2003 commentLet be the set of all complex matrices which are hermitian, that is, , where .
(a) Show that is a real 4-dimensional vector space. Consider the real symmetric bilinear form on this space defined by
Prove that and , where denotes the identity matrix.
(b) Consider the three matrices
Prove that the basis of diagonalizes . Hence or otherwise find the rank and signature of .
(c) Let be the set of all complex matrices which satisfy . Show that is a real 4-dimensional vector space. Given , put
Show that takes values in and is a linear isomorphism between and .
(d) Define a real symmetric bilinear form on by setting , . Show that for all . Find the rank and signature of the symmetric bilinear form defined on .
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3.I.9G
Part IB, 2003 commentLet be a binary quadratic form with integer coefficients. Explain what is meant by the discriminant of . State a necessary and sufficient condition for some form of discriminant to represent an odd prime number . Using this result or otherwise, find the primes which can be represented by the form .
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3.II.19G
Part IB, 2003 commentLet be a finite-dimensional real vector space endowed with a positive definite inner product. A linear map is said to be an orthogonal projection if is self-adjoint and .
(a) Prove that for every orthogonal projection there is an orthogonal decomposition
(b) Let be a linear map. Show that if and , where is the adjoint of , then is an orthogonal projection. [You may find it useful to prove first that if , then and have the same kernel.]
(c) Show that given a subspace of there exists a unique orthogonal projection such that . If and are two subspaces with corresponding orthogonal projections and , show that if and only if is orthogonal to .
(d) Let be a linear map satisfying . Prove that one can define a positive definite inner product on such that becomes an orthogonal projection.
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1.I
Part IB, 2003 commentA particle of mass is confined inside a one-dimensional box of length . Determine the possible energy eigenvalues.
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1.II.18A
Part IB, 2003 commentWhat is the significance of the expectation value
of an observable in the normalized state ? Let and be two observables. By considering the norm of for real values of , show that
The uncertainty of in the state is defined as
Deduce the generalized uncertainty relation,
A particle of mass moves in one dimension under the influence of the potential . By considering the commutator , show that the expectation value of the Hamiltonian satisfies
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2.I.9A
Part IB, 2003 commentWhat is meant by the statement than an operator is hermitian?
A particle of mass moves in the real potential in one dimension. Show that the Hamiltonian of the system is hermitian.
Show that
where is the momentum operator and denotes the expectation value of the operator .
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2.II.18A
Part IB, 2003 commentA particle of mass and energy moving in one dimension is incident from the left on a potential barrier given by
with .
In the limit with held fixed, show that the transmission probability is
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3.II.20A
Part IB, 2003 commentThe radial wavefunction for the hydrogen atom satisfies the equation
Explain the origin of each term in this equation.
The wavefunctions for the ground state and first radially excited state, both with , can be written as
respectively, where and are normalization constants. Determine and the corresponding energy eigenvalues and .
A hydrogen atom is in the first radially excited state. It makes the transition to the ground state, emitting a photon. What is the frequency of the emitted photon?
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3.I.10A
Part IB, 2003 commentWhat are the momentum and energy of a photon of wavelength ?
A photon of wavelength is incident on an electron. After the collision, the photon has wavelength . Show that
where is the scattering angle and is the electron rest mass.
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4.I.9A
Part IB, 2003 commentProve that the two-dimensional Lorentz transformation can be written in the form
where . Hence, show that
Given that frame has speed with respect to and has speed with respect to , use this formalism to find the speed of with respect to .
[Hint: rotation through a hyperbolic angle , followed by rotation through , is equivalent to rotation through .]
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4.II.18A
Part IB, 2003 commentA pion of rest mass decays at rest into a muon of rest mass and a neutrino of zero rest mass. What is the speed of the muon?
In the pion rest frame , the muon moves in the -direction. A moving observer, in a frame with axes parallel to those in the pion rest frame, wishes to take measurements of the decay along the -axis, and notes that the pion has speed with respect to the -axis. Write down the four-dimensional Lorentz transformation relating to and determine the momentum of the muon in . Hence show that in the direction of motion of the muon makes an angle with respect to the -axis, where
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Part IB, 2003
commentDerive the least squares estimators and for the coefficients of the simple linear regression model
where are given constants, , and are independent with .
A manufacturer of optical equipment has the following data on the unit cost (in pounds) of certain custom-made lenses and the number of units made in each order:
\begin{tabular}{l|ccccc} No. of units, & 1 & 3 & 5 & 10 & 12 \ \hline Cost per unit, & 58 & 55 & 40 & 37 & 22 \end{tabular}
Assuming that the conditions underlying simple linear regression analysis are met, estimate the regression coefficients and use the estimated regression equation to predict the unit cost in an order for 8 of these lenses.
[Hint: for the data above, .]
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1.II.12H
Part IB, 2003 commentSuppose that six observations are selected at random from a normal distribution for which both the mean and the variance are unknown, and it is found that , where . Suppose also that 21 observations are selected at random from another normal distribution for which both the mean and the variance are unknown, and it is found that . Derive carefully the likelihood ratio test of the hypothesis against and apply it to the data above at the level.

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2.I.3H
Part IB, 2003 commentLet be a random sample from the distribution, and suppose that the prior distribution for is , where are known. Determine the posterior distribution for , given , and the best point estimate of under both quadratic and absolute error loss.
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2.II.12H
Part IB, 2003 commentAn examination was given to 500 high-school students in each of two large cities, and their grades were recorded as low, medium, or high. The results are given in the table below.
\begin{tabular}{l|ccc} & Low & Medium & High \ \hline City A & 103 & 145 & 252 \ City B & 140 & 136 & 224 \end{tabular}
Derive carefully the test of homogeneity and test the hypothesis that the distributions of scores among students in the two cities are the same.

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4.I.3H
Part IB, 2003 commentThe following table contains a distribution obtained in 320 tosses of 6 coins and the corresponding expected frequencies calculated with the formula for the binomial distribution for and .
\begin{tabular}{l|rrrrrrr} No. heads & 0 & 1 & 2 & 3 & 4 & 5 & 6 \ \hline Observed frequencies & 3 & 21 & 85 & 110 & 62 & 32 & 7 \ Expected frequencies & 5 & 30 & 75 & 100 & 75 & 30 & 5 \end{tabular}
Conduct a goodness-of-fit test at the level for the null hypothesis that the coins are all fair.
[Hint:
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4.II.12H
Part IB, 2003 commentState and prove the Rao-Blackwell theorem.
Suppose that are independent random variables uniformly distributed over . Find a two-dimensional sufficient statistic for . Show that an unbiased estimator of is .
Find an unbiased estimator of which is a function of and whose mean square error is no more than that of .
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A2.7
Part II, 2003 comment(i) What are geodesic polar coordinates at a point on a surface with a Riemannian metric ?
Assume that
for geodesic polar coordinates and some function . What can you say about and at ?
(ii) Given that the Gaussian curvature may be computed by the formula , show that for small the area of the geodesic disc of radius centred at is
where is a function satisfying .
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A3.7
Part II, 2003 comment(i) Suppose that is a curve in the Euclidean -plane and that is parameterized by its arc length . Suppose that in Euclidean is the surface of revolution obtained by rotating about the -axis. Take as coordinates on , where is the angle of rotation.
Show that the Riemannian metric on induced from the Euclidean metric on is
(ii) For the surface described in Part (i), let and . Show that, along any geodesic on , the quantity is constant. Here is the metric tensor on .
[You may wish to compute for any vector field , where are functions of . Then use symmetry to compute , which is the rate of change of along .]
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A4.7
Part II, 2003 commentWrite an essay on the Theorema Egregium for surfaces in .
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A1.8
Part II, 2003 comment(i) State Brooks' Theorem, and prove it in the case of a 3 -connected graph.
(ii) Let be a bipartite graph, with vertex classes and , each of order . If contains no cycle of length 4 show that
For which integers are there examples where equality holds?
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A ,
Part II, 2003 comment(i) State and prove a result of Euler relating the number of vertices, edges and faces of a plane graph. Use this result to exhibit a non-planar graph.
(ii) State the vertex form of Menger's Theorem and explain how it follows from an appropriate version of the Max-flow-min-cut Theorem. Let . Show that every -connected graph of order at least contains a cycle of length at least .
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A4.9
Part II, 2003 commentWrite an essay on the vertex-colouring of graphs drawn on compact surfaces other than the sphere. You should include a proof of Heawood's bound, and an example of a surface for which this bound is not attained.
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A1.9
Part II, 2003 comment(i) Let be an odd prime and a strictly positive integer. Prove that the multiplicative group of relatively prime residue classes modulo is cyclic.
[You may assume that the result is true for .]
(ii) Let , where and are distinct odd primes. Let denote the set of all integers which are relatively prime to . We recall that is said to be an Euler pseudo-prime to the base if
If is an Euler pseudo-prime to the base , but is not an Euler pseudo-prime to the base , prove that is not an Euler pseudo-prime to the base . Let denote any of the primes . Prove that there exists a such that
and deduce that is not an Euler pseudo-prime to this base . Hence prove that is not an Euler pseudo-prime to the base for at least half of all the relatively prime residue classes .
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A3.9
Part II, 2003 comment(i) Let be a real number and let , where the product is taken over all primes . Prove that .
(ii) Define the continued fraction of any positive irrational real number . Illustrate your definition by computing the continued fraction of .
Suppose that are positive integers with and that has the periodic continued fraction . Prove that .
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A4.10
Part II, 2003 commentWrite an essay describing the factor base method for factorising a large odd positive integer . Your essay should include a detailed explanation of how the continued fraction of can be used to find a suitable factor base.
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A2.9
Part II, 2003 comment(i) Answer the following questions briefly but clearly.
(a) How does coding theory apply when the error rate ?
(b) Give an example of a code which is not a linear code.
(c) Give an example of a linear code which is not a cyclic code.
(d) Give an example of a general feedback register with output , and initial fill , such that
for all .
(e) Explain why the original Hamming code can not always correct two errors.
(ii) Describe the Rabin-Williams scheme for coding a message as modulo a certain . Show that, if is chosen appropriately, breaking this code is equivalent to factorising the product of two primes.
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A2.10
Part II, 2003 comment(i) Consider a network with node set and set of directed arcs equipped with functions and with . Given we define the differential by for . We say that is a feasible differential if
Write down a necessary and sufficient condition on for the existence of a feasible differential and prove its necessity.
Assuming Minty's Lemma, describe an algorithm to construct a feasible differential and outline how this algorithm establishes the sufficiency of the condition you have given.
(ii) Let , where are finite sets. A matching in is a subset such that, for all and ,
A matching is maximal if for any other matching with we must have . Formulate the problem of finding a maximal matching in in terms of an optimal distribution problem on a suitably defined network, and hence in terms of a standard linear optimization problem.
[You may assume that the optimal distribution subject to integer constraints is integervalued.]
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A3.10
Part II, 2003 comment(i) Consider the problem
where and . State and prove the Lagrangian sufficiency theorem.
In each of the following cases, where and , determine whether the Lagrangian sufficiency theorem can be applied to solve the problem:
(ii) Consider the problem in
where is a positive-definite symmetric matrix, is an matrix, and . Explain how to reduce this problem to the solution of simultaneous linear equations.
Consider now the problem
Describe the active set method for its solution.
Consider the problem
where . Draw a diagram partitioning the -plane into regions according to which constraints are active at the minimum.
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A1.10
Part II, 2003 comment(i) We work over the field of two elements. Define what is meant by a linear code of length . What is meant by a generator matrix for a linear code?
Define what is meant by a parity check code of length . Show that a code is linear if and only if it is a parity check code.
Give the original Hamming code in terms of parity checks and then find a generator matrix for it.
[You may use results from the theory of vector spaces provided that you quote them correctly.]
(ii) Suppose that and let be the largest information rate of any binary error correcting code of length which can correct errors.
Show that
where
[You may assume any form of Stirling's theorem provided that you quote it correctly.]
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A4.11
Part II, 2003 commentDefine the optimal distribution problem. State what it means for a circuit to be flow-augmenting, and what it means for to be unbalanced. State the optimality theorem for flows. Describe the simplex-on-a-graph algorithm, giving a brief justification of its stopping rules.
Consider the problem of finding, in the network shown below, a minimum-cost flow from to of value 2 . Here the circled numbers are the upper arc capacities, the lower arc capacities all being zero, and the uncircled numbers are costs. Apply the simplex-on-agraph algorithm to solve this problem, taking as initial flow the superposition of a unit flow along the path and a unit flow along the path .

Part II 2003
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A1.13
Part II, 2003 comment(i) Suppose , are independent binomial observations, with , , where are known, and we wish to fit the model
where are given covariates, each of dimension . Let be the maximum likelihood estimators of . Derive equations for and state without proof the form of the approximate distribution of .
(ii) In 1975 , data were collected on the 3-year survival status of patients suffering from a type of cancer, yielding the following table
\begin{tabular}{ccrr} & & \multicolumn{2}{c}{ survive? } \ age in years & malignant & yes & no \ under 50 & no & 77 & 10 \ under 50 & yes & 51 & 13 \ & no & 51 & 11 \ & yes & 38 & 20 \ & no & 7 & 3 \ & yes & 6 & 3 \end{tabular}
Here the second column represents whether the initial tumour was not malignant or was malignant.
Let be the number surviving, for age group and malignancy status , for and , and let be the corresponding total number. Thus , . Assume . The results from fitting the model
with give , and deviance . What do you conclude?
Why do we take in the model?
What "residuals" should you compute, and to which distribution would you refer them?
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A2.12
Part II, 2003 comment(i) Suppose are independent Poisson variables, and
where are two unknown parameters, and are given covariates, each of dimension 1. Find equations for , the maximum likelihood estimators of , and show how an estimate of may be derived, quoting any standard theorems you may need.
(ii) By 31 December 2001, the number of new vCJD patients, classified by reported calendar year of onset, were
for the years
Discuss carefully the (slightly edited) output for these data given below, quoting any standard theorems you may need.
year
year
[1] 1994199519961997199819992000
tot
[1]
first.glm - glm(tot year, family = poisson)
(first.glm)
Call:
glm(formula tot year, family poisson
Coefficients
Estimate Std. Error z value
(Intercept)
year
(Dispersion parameter for poisson family taken to be 1)
Null deviance: on 6 degrees of freedom
Residual deviance: on 5 degrees of freedom
Number of Fisher Scoring iterations: 3
Part II 2003
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A4.14
Part II, 2003 commentThe nave height , and the nave length for 16 Gothic-style cathedrals and 9 Romanesque-style cathedrals, all in England, have been recorded, and the corresponding output (slightly edited) is given below.

You may assume that are in suitable units, and that "style" has been set up as a factor with levels 1,2 corresponding to Gothic, Romanesque respectively.
(a) Explain carefully, with suitable graph(s) if necessary, the results of this analysis.
(b) Using the general model (in the conventional notation) explain carefully the theory needed for (a).
[Standard theorems need not be proved.]
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A1.14
Part II, 2003 comment(i) An electron of mass and spin moves freely inside a cubical box of side . Verify that the energy eigenstates of the system are where the spatial wavefunction is given by
and
Give the corresponding energy eigenvalues.
A second electron is inserted into the box. Explain how the Pauli principle determines the structure of the wavefunctions associated with the lowest energy level and the first excited energy level. What are the values of the energy in these two levels and what are the corresponding degeneracies?
(ii) When the side of the box, , is large, the number of eigenstates available to the electron with energy in the range is . Show that
A large number, , of electrons are inserted into the box. Explain how the ground state is constructed and define the Fermi energy, . Show that in the ground state
When a magnetic field in the -direction is applied to the system, an electron with spin up acquires an additional energy and an electron with spin down an energy , where is the magnetic moment of the electron and . Describe, for the case , the structure of the ground state of the system of electrons in the box and show that
Calculate the induced magnetic moment, , of the ground state of the system and show that for a weak magnetic field the magnetic moment is given by
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A2.14
Part II, 2003 comment(i) A system of distinguishable non-interacting particles has energy levels with degeneracy , for each particle. Show that in thermal equilibrium the number of particles with energy is given by
where and are parameters whose physical significance should be briefly explained.
A gas comprises a set of atoms with non-degenerate energy levels . Assume that the gas is dilute and the motion of the atoms can be neglected. For such a gas the atoms can be treated as distinguishable. Show that when the system is at temperature , the number of atoms at level and the number at level satisfy
where is Boltzmann's constant.
(ii) A system of bosons has a set of energy levels with degeneracy , for each particle. In thermal equilibrium at temperature the number of particles in level is
What is the value of when the particles are photons?
Given that the density of states for photons of frequency in a cubical box of side is
where is the speed of light, show that at temperature the density of photons in the frequency range is where
Deduce the energy density, , for photons of frequency .
The cubical box is occupied by the gas of atoms described in Part (i) in the presence of photons at temperature . Consider the two atomic levels and where and . The rate of spontaneous photon emission for the transition is . The rate of absorption is and the rate of stimulated emission is . Show that the requirement that these processes maintain the atoms and photons in thermal equilibrium yields the relations
and
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A4.16
Part II, 2003 commentDescribe the energy band structure available to electrons moving in crystalline materials. How can it be used to explain the properties of crystalline materials that are conductors, insulators and semiconductors?
Where does the Fermi energy lie in an intrinsic semiconductor? Describe the process of doping of semiconductors and explain the difference between -type and -type doping. What is the effect of the doping on the position of the Fermi energy in the two cases?
Why is there a potential difference across a junction of -type and -type semiconductors?
Derive the relation
between the current, , and the voltage, , across an junction, where is the total minority current in the semiconductor and is the charge on the electron, is the temperature and is Boltzmann's constant. Your derivation should include an explanation of the terms majority current and minority current.
Why can the junction act as a rectifier?
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A1.16
Part II, 2003 comment(i) Explain briefly how the relative motion of galaxies in a homogeneous and isotropic universe is described in terms of the scale factor (where is time). In particular, show that the relative velocity of two galaxies is given in terms of their relative displacement by the formula , where is a function that you should determine in terms of . Given that , obtain a formula for the distance to the cosmological horizon at time . Given further that , for and constant , compute . Hence show that as .
(ii) A homogeneous and isotropic model universe has energy density and pressure , where is the speed of light. The evolution of its scale factor is governed by the Friedmann equation
where the overdot indicates differentiation with respect to . Use the "Fluid" equation
to obtain an equation for the acceleration . Assuming and , show that cannot increase with time as long as , nor decrease if . Hence determine the late time behaviour of for . For show that an initially expanding universe must collapse to a "big crunch" at which . How does behave as ? Given that , determine the form of near the big crunch. Discuss the qualitative late time behaviour for .
Cosmological models are often assumed to have an equation of state of the form for constant . What physical principle requires ? Matter with is called "stiff matter" by cosmologists. Given that , determine for a universe that contains only stiff matter. In our Universe, why would you expect stiff matter to be negligible now even if it were significant in the early Universe?
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A3.14
Part II, 2003 comment(i) The pressure and mass density , at distance from the centre of a spherically-symmetric star, obey the pressure-support equation
where , and the prime indicates differentiation with respect to . Let be the total volume of the star, and its average pressure. Use the pressure-support equation to derive the "virial theorem"
where is the total gravitational potential energy [Hint: multiply by ]. If a star is assumed to be a self-gravitating ball of a non-relativistic ideal gas then it can be shown that
where is the total kinetic energy. Use this result to show that the total energy is negative. When nuclear reactions have converted the hydrogen in a star's core to helium the core contracts until the helium is converted to heavier elements, thereby increasing the total energy of the star. Explain briefly why this converts the star into a "Red Giant". (ii) Write down the first law of thermodynamics for the change in energy of a system at temperature , pressure and chemical potential as a result of small changes in the entropy , volume and particle number . Use this to show that
The microcanonical ensemble is the set of all accessible microstates of a system at fixed . Define the canonical and grand-canonical ensembles. Why are the properties of a macroscopic system independent of the choice of thermodynamic ensemble?
The Gibbs "grand potential" can be defined as
Use the first law to find expressions for as partial derivatives of . A system with variable particle number has non-degenerate energy eigenstates labeled by , for each , with energy eigenvalues . If the system is in equilibrium at temperature and chemical potential then the probability that it will be found in a particular -particle state is given by the Gibbs probability distribution
where is Boltzmann's constant. Deduce an expression for the normalization factor as a function of and , and hence find expressions for the partial derivatives
in terms of .
Why does also depend on the volume ? Given that a change in at fixed leaves unchanged the Gibbs probability distribution, deduce that
Use your results to show that
for some constant .
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A4.18
Part II, 2003 commentLet be the density of states of a particle in volume as a function of the magnitude of the particle's momentum. Explain why , where is Planck's constant. Write down the Bose-Einstein and Fermi-Dirac distributions for the (average) number of particles of an ideal gas with momentum . Hence write down integrals for the (average) total number of particles and the (average) total energy as functions of temperature and chemical potential . Why do and also depend on the volume
Electromagnetic radiation in thermal equilibrium can be regarded as a gas of photons. Why are photons "ultra-relativistic" and how is photon momentum related to the frequency of the radiation? Why does a photon gas have zero chemical potential? Use your formula for to express the energy density of electromagnetic radiation in the form
where is a function of that you should determine up to a dimensionless multiplicative constant. Show that is independent of when , where is Boltzmann's constant. Let be the value of at the maximum of the function ; how does depend on ?
Let be the photon number density at temperature . Show that for some power , which you should determine. Why is unchanged as the volume is increased quasi-statically? How does depend on under these circumstances? Applying your result to the Cosmic Microwave Background Radiation (CMBR), deduce how the temperature of the CMBR depends on the scale factor of the Universe. At a time when , the Universe underwent a transition from an earlier time at which it was opaque to a later time at which it was transparent. Explain briefly the reason for this transition and its relevance to the CMBR.
An ideal gas of fermions of mass is in equilibrium at temperature and chemical potential with a gas of its own anti-particles and photons . Assuming that chemical equilibrium is maintained by the reaction
determine the chemical potential of the antiparticles. Let and be the number densities of and , respectively. What will their values be for if ? Given that , but , show that
where is the fermion number density at zero chemical potential and is a positive function of the dimensionless ratio . What is when ?
Given that , obtain an expression for the ratio in terms of and the function . Supposing that is either a proton or neutron, why should you expect the ratio to remain constant as the Universe expands?
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A1.17
Part II, 2003 comment(i) Define the character of a representation of a finite group . Show that if and only if is irreducible, where
If and , what are the possible dimensions of the representation
(ii) State and prove Schur's first and second lemmas.
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A3.15
Part II, 2003 comment(i) Given that the character of an transformation in the -dimensional irreducible representation is given by
show how the direct product representation decomposes into irreducible representations.
(ii) Find the decomposition of the direct product representation of into irreducible representations.
Mesons consist of one quark and one antiquark. The scalar Meson Octet consists of the following particles: , and .
Use the direct product representation of to identify the quark-type of the particles in the scalar Meson Octet. Deduce the quark-type of the singlet state contained in .
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A1.18
Part II, 2003 comment(i) A solute occupying a domain has concentration and is created at a rate per unit volume; is the flux of solute per unit area; are position and time. Derive the transport equation
State Fick's Law of diffusion and hence write down the diffusion equation for for a case in which the solute flux occurs solely by diffusion, with diffusivity .
In a finite domain and the steady-state distribution of depend only on is equal to at and at . Find in the following two cases: (a) , (b) ,
where and are positive constants.
Show that there is no steady solution satisfying the boundary conditions if
(ii) For the problem of Part (i), consider the case , where and are positive constants. Calculate the steady-state solution, , assuming that for any integer .
Now let
where . Find the equations, boundary and initial conditions satisfied by . Solve the problem using separation of variables and show that
for some constants . Write down an integral expression for , show that
and comment on the behaviour of the solution for large times in the two cases and .
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A3.16
Part II, 2003 comment(i) When a solid crystal grows into a supercooled infinite melt, latent heat must be removed from the interface by diffusion into the melt. Write down the equation and boundary conditions satisfied by the temperature in the melt, where is position and time, in terms of the following material properties: solid density , specific heat capacity , coefficient of latent heat per unit mass , thermal conductivity , melting temperature . You may assume that the densities of the melt and the solid are the same and that temperature in the melt far from the interface is , where is a positive constant.
A spherical crystal of radius grows into such a melt with . Use dimensional analysis to show that is proportional to .
(ii) Show that the above problem should have a similarity solution of the form
where is the radial coordinate in spherical polars and is the thermal diffusivity. Recalling that, for spherically symmetric , write down the equation and boundary conditions to be satisfied by . Hence show that the radius of the crystal is given by , where satisfies the equation
and .
Integrate the left hand side of this equation by parts, to give
Hence show that a solution with small must have , which is self-consistent if is large.
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A4.19
Part II, 2003 commentA shallow layer of fluid of viscosity , density and depth lies on a rigid horizontal plane and is bounded by impermeable barriers at and . Gravity acts vertically and a wind above the layer causes a shear stress to be exerted on the upper surface in the direction. Surface tension is negligible compared to gravity.
(a) Assuming that the steady flow in the layer can be analysed using lubrication theory, show that the horizontal pressure gradient is given by and hence that
Show also that the fluid velocity at the surface is equal to , and sketch the velocity profile for .
(b) In the case in which is a constant, , and assuming that the difference between and its average value remains small compared with , show that
provided that
(c) Surfactant at surface concentration is added to the surface, so that now
where is a positive constant. The surfactant is advected by the surface fluid velocity and also experiences a surface diffusion with diffusivity . Write down the equation for conservation of surfactant, and hence show that
From equations (1), (2) and (3) deduce that
where is a constant. Assuming once more that , and that at , show further that
provided that
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A1.19
Part II, 2003 comment(i) Explain the concepts of: traction on an element of surface; the stress tensor; the strain tensor in an elastic medium. Derive a relationship between the two tensors for a linear isotropic elastic medium, stating clearly any assumption you need to make.
(ii) State what is meant by an wave in a homogeneous isotropic elastic medium. An SH wave in a medium with shear modulus and density is incident at angle on an interface with a medium with shear modulus and density . Evaluate the form and amplitude of the reflected wave and transmitted wave. Comment on the case , where and .
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A2.16
Part II, 2003 comment(i) Explain briefly what is meant by the concepts of hydrostatic equilibrium and the buoyancy frequency. Evaluate an expression for the buoyancy frequency in an incompressible inviscid fluid with stable density profile .
(ii) Explain briefly what is meant by the Boussinesq approximation.
Write down the equations describing motions of small amplitude in an incompressible, stratified, Boussinesq fluid of constant buoyancy frequency.
Derive the resulting dispersion relationship for plane wave motion. Show that there is a maximum frequency for the waves and explain briefly why this is the case.
What would be the response to a solid body oscillating at a frequency in excess of the maximum?
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A4.20
Part II, 2003 commentDefine the Rossby number. Under what conditions will a fluid flow be at (a) high and (b) low values of the Rossby number? Briefly describe both an oceanographic and a meteorological example of each type of flow.
Explain the concept of quasi-geostrophy for a thin layer of homogeneous fluid in a rapidly rotating system. Write down the quasi-geostrophic approximation for the vorticity in terms of the pressure, the fluid density and the rate of rotation. Define the potential vorticity and state the associated conservation law.
A broad current flows directly eastwards ( direction) with uniform velocity across a flat ocean basin of depth . The current encounters a low, two-dimensional ridge of width and height , whose axis is aligned in the north-south direction. Neglecting any effects of stratification and assuming a constant vertical rate of rotation , such that the Rossby number is small, determine the effect of the ridge on the current. Show that the direction of the current after it leaves the ridge is dependent on the cross-sectional area of the ridge, but not on the explicit form of .
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A2.17
Part II, 2003 comment(i) Explain how to solve the Fredholm integral equation of the second kind,
in the case where is of the separable (degenerate) form
(ii) For what values of the real constants and does the equation
have (a) a unique solution, (b) no solution?
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A3.17
Part II, 2003 comment(i) Explain what is meant by the assertion: "the series is asymptotic to as .
Consider the integral
where is real and has the asymptotic expansion
as , with . State Watson's lemma describing the asymptotic behaviour of as , and determine an expression for the general term in the asymptotic series.
(ii) Let
for . Show that
as .
Suggest, for the case that is smaller than unity, the point at which this asymptotic series should be truncated so as to produce optimal numerical accuracy.
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A4.21
Part II, 2003 commentLet denote the solution for of
subject to the conditions that and as , where ; it may be assumed that for . Write in the form
and consider an asymptotic expansion of the form
valid in the limit with . Find and .
It is known that the solution is of the form
where
and the constant factor depends on . By letting , show that the expression
satisfies the relevant differential equation with an error of as . Comment on the relationship between your answers for and .
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A2.18
Part II, 2003 comment(i) Write down the shock condition associated with the equation
where . Discuss briefly two possible heuristic approaches to justifying this shock condition.
(ii) According to shallow water theory, waves on a uniformly sloping beach are described by the equations
where is the constant slope of the beach, is the gravitational acceleration, is the fluid velocity, and is the elevation of the fluid surface above the undisturbed level.
Find the characteristic velocities and the characteristic form of the equations.
What are the Riemann variables and how do they vary with on the characteristics?
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A3.18
Part II, 2003 comment(i) Write down a Lax pair for the equation
Discuss briefly, without giving mathematical details, how this pair can be used to solve the Cauchy problem on the infinite line for this equation. Discuss how this approach can be used to solve the analogous problem for the nonlinear Schrödinger equation.
(ii) Let satisfy the equations
where is a constant.
(a) Show that the above equations are compatible provided that both satisfy the Sine-Gordon equation
(b) Use the above result together with the fact that
to show that the one-soliton solution of the Sine-Gordon equation is given by
where is a constant.
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A4.22
Part II, 2003 commentLet denote the boundary values of functions which are analytic inside and outside a disc of radius centred at the origin. Let denote the boundary of this disc.
Suppose that satisfy the jump condition
(a) Show that the associated index is 1 .
(b) Find the canonical solution of the homogeneous problem, i.e. the solution satisfying
(c) Find the general solution of the Riemann-Hilbert problem satisfying the above jump condition as well as
(d) Use the above result to solve the linear singular integral problem
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A1.1 B1.1
Part II, 2003 comment(i) Let be a simple symmetric random walk in , starting from , and set . Determine the quantities and and .
(ii) Let be a discrete-time Markov chain with state-space and transition matrix . What does it mean to say that a state is recurrent? Prove that is recurrent if and only if , where denotes the entry in .
Show that the simple symmetric random walk in is recurrent.
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A2.1
Part II, 2003 comment(i) What is meant by a Poisson process of rate ? Show that if and are independent Poisson processes of rates and respectively, then is also a Poisson process, and determine its rate.
(ii) A Poisson process of rate is observed by someone who believes that the first holding time is longer than all subsequent holding times. How long on average will it take before the observer is proved wrong?
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A3.1 B3.1
Part II, 2003 comment(i) Consider the continuous-time Markov chain with state-space and -matrix
Set
and
Determine which, if any, of the processes and are Markov chains.
(ii) Find an invariant distribution for the chain given in Part (i). Suppose . Find, for all , the probability that .
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A4.1
Part II, 2003 commentConsider a pack of cards labelled . We repeatedly take the top card and insert it uniformly at random in one of the 52 possible places, that is, either on the top or on the bottom or in one of the 50 places inside the pack. How long on average will it take for the bottom card to reach the top?
Let denote the probability that after iterations the cards are found in increasing order. Show that, irrespective of the initial ordering, converges as , and determine the limit . You should give precise statements of any general results to which you appeal.
Show that, at least until the bottom card reaches the top, the ordering of cards inserted beneath it is uniformly random. Hence or otherwise show that, for all ,
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A1.2 B1.2
Part II, 2003 comment(i) Consider particles moving in 3 dimensions. The Cartesian coordinates of these particles are . Now consider an invertible change of coordinates to coordinates , so that one may express as . Show that the velocity of the system in Cartesian coordinates is given by the following expression:
Furthermore, show that Lagrange's equations in the two coordinate systems are related via
(ii) Now consider the case where there are constraints applied, . By considering the , and a set of independent coordinates , as a set of new coordinates, show that the Lagrange equations of the constrained system, i.e.
(where the are Lagrange multipliers) imply Lagrange's equations for the unconstrained coordinates, i.e.
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A2.2 B2.1
Part II, 2003 comment(i) The trajectory of a non-relativistic particle of mass and charge moving in an electromagnetic field obeys the Lorentz equation
Show that this equation follows from the Lagrangian
where is the electromagnetic scalar potential and the vector potential, so that
(ii) Let . Consider a particle moving in a constant magnetic field which points in the direction. Show that the particle moves in a helix about an axis pointing in the direction. Evaluate the radius of the helix.
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A3.2
Part II, 2003 comment(i) An axisymmetric bowling ball of mass has the shape of a sphere of radius . However, it is biased so that the centre of mass is located a distance away from the centre, along the symmetry axis.
The three principal moments of inertia about the centre of mass are . The ball starts out in a stable equilibrium at rest on a perfectly frictionless flat surface with the symmetry axis vertical. The symmetry axis is then tilted through , the ball is spun about this axis with an angular velocity , and the ball is released.
Explain why the centre of mass of the ball moves only in the vertical direction during the subsequent motion. Write down the Lagrangian for the ball in terms of the usual Euler angles and .
(ii) Show that there are three independent constants of the motion. Eliminate two of the angles from the Lagrangian and find the effective Lagrangian for the coordinate .
Find the maximum and minimum values of in the motion of the ball when the quantity is (a) very small and (b) very large.
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A4.2
Part II, 2003 commentThe action of a Hamiltonian system may be regarded as a function of the final coordinates , and the final time by setting
where the initial coordinates and time are held fixed, and are the solutions to Hamilton's equations with Hamiltonian , satisfying .
(a) Show that under an infinitesimal change of the final coordinates and time , the change in is
(b) Hence derive the Hamilton-Jacobi equation
(c) If we can find a solution to ,
where are integration constants, then we can use as a generating function of type , where
Show that the Hamiltonian in the new coordinates vanishes.
(d) Write down and solve the Hamilton-Jacobi equation for the one-dimensional simple harmonic oscillator, where . Show the solution takes the form . Using this as a generating function show that the new coordinates are constants of the motion and give their physical interpretation.
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A1.3
Part II, 2003 comment(i) Let be a continuous linear map between two Hilbert spaces . Define the adjoint of . Explain what it means to say that is Hermitian or unitary.
Let be a bounded continuous function. Show that the map
with is a continuous linear map and find its adjoint. When is Hermitian? When is it unitary?
(ii) Let be a closed, non-empty, convex subset of a real Hilbert space . Show that there exists a unique point with minimal norm. Show that is characterised by the property
Does this result still hold when is not closed or when is not convex? Justify your answers.
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A2.3 B2.2
Part II, 2003 comment(i) Define the dual of a normed vector space . Show that the dual is always a complete normed space.
Prove that the vector space , consisting of those real sequences for which the norm
is finite, has the vector space of all bounded sequences as its dual.
(ii) State the Stone-Weierstrass approximation theorem.
Let be a compact subset of . Show that every can be uniformly approximated by a sequence of polynomials in variables.
Let be a continuous function on . Deduce that
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A3.3 B3.2
Part II, 2003 comment(i) Let be a point of the compact interval and let be defined by . Show that
is a continuous, linear map but that
is not continuous.
(ii) Consider the space of -times continuously differentiable functions on the interval . Write
for . Show that is a complete normed space. Is the space also complete?
Let be an -times continuously differentiable map and define
Show that is a continuous linear map when is equipped with the norm .
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A4.3
Part II, 2003 comment(i) State the Monotone Convergence Theorem and explain briefly how to prove it.
(ii) For which real values of is ?
Let . Using the Monotone Convergence Theorem and the identity
prove carefully that
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A1.4
Part II, 2003 comment(i) Let be a prime number. Show that a group of order has a nontrivial normal subgroup, that is, is not a simple group.
(ii) Let and be primes, . Show that a group of order has a normal Sylow -subgroup. If has also a normal Sylow -subgroup, show that is cyclic. Give a necessary and sufficient condition on and for the existence of a non-abelian group of order . Justify your answer.
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B1.3
Part II, 2003 comment(i) Let be a prime number. Show that a group of order has a nontrivial normal subgroup, that is, is not a simple group.
(ii) Let and be primes, . Show that a group of order has a normal Sylow -subgroup. If has also a normal Sylow -subgroup, show that is cyclic. Give a necessary and sufficient condition on and for the existence of a non-abelian group of order . Justify your answer.
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A3.4
Part II, 2003 comment(i) Let be the splitting field of the polynomial over the rationals. Find the Galois group of and describe its action on the roots of .
(ii) Let be the splitting field of the polynomial (where ) over the rationals. Assuming that the polynomial is irreducible, prove that the Galois group of the extension is either , or , or the dihedral group .
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A2.4 B2.3
Part II, 2003 comment(i) In each of the following two cases, determine a highest common factor in :
(a) ;
(b) .
(ii) State and prove the Eisenstein criterion for irreducibility of polynomials with integer coefficients. Show that, if is prime, the polynomial
is irreducible over .
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A4.4
Part II, 2003 commentWrite an essay on the theory of invariants. Your essay should discuss the theorem on the finite generation of the ring of invariants, the theorem on elementary symmetric functions, and some examples of calculation of rings of invariants.
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A B
Part II, 2003 comment(i) Using Maxwell's equations as they apply to magnetostatics, show that the magnetic field can be described in terms of a vector potential on which the condition may be imposed. Hence derive an expression, valid at any point in space, for the vector potential due to a steady current distribution of density that is non-zero only within a finite domain.
(ii) Verify that the vector potential that you found in Part (i) satisfies , and use it to obtain the Biot-Savart law expression for . What is the corresponding result for a steady surface current distribution of density ?
In cylindrical polar coordinates (oriented so that ) a surface current
flows in the plane . Given that
show that the magnetic field at the point has -component
State, with justification, the full result for at the point .
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A2
Part II, 2003 comment(i) A plane electromagnetic wave has electric and magnetic fields
for constant vectors , constant positive angular frequency and constant wavevector . Write down the vacuum Maxwell equations and show that they imply
Show also that , where is the speed of light.
(ii) State the boundary conditions on and at the surface of a perfect conductor. Let be the surface charge density and s the surface current density on . How are and related to and ?
A plane electromagnetic wave is incident from the half-space upon the surface of a perfectly conducting medium in . Given that the electric and magnetic fields of the incident wave take the form with
and
find .
Reflection of the incident wave at produces a reflected wave with electric field
with
By considering the boundary conditions at on the total electric field, show that
Show further that the electric charge density on the surface takes the form
for a constant that you should determine. Find the magnetic field of the reflected wave and hence the surface current density on the surface .
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A3.5 B3.3
Part II, 2003 comment(i) Given the electric field (in cartesian components)
use the Maxwell equation
to find subject to the boundary condition that as .
Let be the planar rectangular surface in the -plane with corners at
where is a constant and is some function of time. The magnetic flux through is given by the surface integral
Compute as a function of .
Let be the closed rectangular curve that bounds the surface , taken anticlockwise in the -plane, and let be its velocity (which depends, in this case, on the segment of being considered). Compute the line integral
Hence verify that
(ii) A surface is bounded by a time-dependent closed curve such that in time it sweeps out a volume . By considering the volume integral
and using the divergence theorem, show that the Maxwell equation implies that
where is the magnetic flux through as given in Part (i). Hence show, using (1) and Stokes' theorem, that (2) is a consequence of Maxwell's equations.
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A4.5
Part II, 2003 commentLet be the electric field due to a continuous static charge distribution for which as . Starting from consideration of a finite system of point charges, deduce that the electrostatic energy of the charge distribution is
where the volume integral is taken over all space.
A sheet of perfectly conducting material in the form of a surface , with unit normal , carries a surface charge density . Let denote the normal components of the electric field on either side of . Show that
Three concentric spherical shells of perfectly conducting material have radii with . The innermost and outermost shells are held at zero electric potential. The other shell is held at potential . Find the potentials in and in . Compute the surface charge density on the shell of radius . Use the formula to compute the electrostatic energy of the system.
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A1.6
Part II, 2003 comment(i) State and prove Dulac's Criterion for the non-existence of periodic orbits in . Hence show (choosing a weighting factor of the form ) that there are no periodic orbits of the equations
(ii) State the Poincaré-Bendixson Theorem. A model of a chemical reaction (the Brusselator) is defined by the second order system
where are positive parameters. Show that there is a unique fixed point. Show that, for a suitable choice of , trajectories enter the closed region bounded by , and . Deduce that when , the system has a periodic orbit.
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A2.6 B2.4
Part II, 2003 comment(i) What is a Liapunov function?
Consider the second order ODE
By finding a suitable Liapunov function of the form , where and are to be determined, show that the origin is asymptotically stable. Using your form of , find the greatest value of such that a trajectory through is guaranteed to tend to the origin as .
[Any theorems you use need not be proved but should be clearly stated.]
(ii) Explain the use of the stroboscopic method for investigating the dynamics of equations of the form , when . In particular, for , derive the equations, correct to order ,
where the brackets denote an average over the period of the unperturbed oscillator.
Find the form of the right hand sides of these equations explicitly when , where . Show that apart from the origin there is another fixed point of , and determine its stability. Sketch the trajectories in space in the case . What do you deduce about the dynamics of the full equation?
[You may assume that .]
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A3.6 B3.4
Part II, 2003 comment(i) Define the Poincaré index of a curve for a vector field . Explain why the index is uniquely given by the sum of the indices for small curves around each fixed point within . Write down the indices for a saddle point and for a focus (spiral) or node, and show that the index of a periodic solution of has index unity.
A particular system has a periodic orbit containing five fixed points, and two further periodic orbits. Sketch the possible arrangements of these orbits, assuming there are no degeneracies.
(ii) A dynamical system in depending on a parameter has a homoclinic orbit when . Explain how to determine the stability of this orbit, and sketch the different behaviours for and in the case that the orbit is stable.
Now consider the system
where are constants. Show that the origin is a saddle point, and that if there is an orbit homoclinic to the origin then are related by
where the integral is taken round the orbit. Evaluate this integral for small by approximating by its form when . Hence give conditions on (small) that lead to a stable homoclinic orbit at the origin. [Note that .]
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A4.6
Part II, 2003 commentExplain what is meant by a steady-state bifurcation of a fixed point of an ODE , in , where is a real parameter. Give examples for of equations exhibiting saddle-node, transcritical and pitchfork bifurcations.
Consider the system in , with ,
Show that the fixed point has a bifurcation when , while the fixed points have a bifurcation when . By finding the first approximation to the extended centre manifold, construct the normal form at the bifurcation point in each case, and determine the respective bifurcation types. Deduce that for just greater than , and for just less than 1 , there is a stable pair of "mixed-mode" solutions with .
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A B1.12
Part II, 2003 comment(i) State Zorn's Lemma. Use Zorn's Lemma to prove that every real vector space has a basis.
(ii) State the Bourbaki-Witt Theorem, and use it to prove Zorn's Lemma, making clear where in the argument you appeal to the Axiom of Choice.
Conversely, deduce the Bourbaki-Witt Theorem from Zorn's Lemma.
If is a non-empty poset in which every chain has an upper bound, must be chain-complete?
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B2.11
Part II, 2003 commentState the Axiom of Replacement.
Show that for any set there is a transitive set that contains , indicating where in your argument you have used the Axiom of Replacement. No form of recursion theorem may be assumed without proof.
Which of the following are true and which are false? Give proofs or counterexamples as appropriate. You may assume standard properties of ordinals.
(a) If is a transitive set then is an ordinal.
(b) If each member of a set is an ordinal then is an ordinal.
(c) If is a transitive set and each member of is an ordinal then is an ordinal.
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A3.8 B3.11
Part II, 2003 comment(i) What does it mean for a function from to to be recursive? Write down a function that is not recursive. You should include a proof that your example is not recursive.
(ii) What does it mean for a subset of to be recursive, and what does it mean for it to be recursively enumerable? Give, with proof, an example of a set that is recursively enumerable but not recursive. Prove that a set is recursive if and only if both it and its complement are recursively enumerable. If a set is recursively enumerable, must its complement be recursively enumerable?
[You may assume the existence of any universal recursive functions or universal register machine programs that you wish.]
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A4.8 B4.10
Part II, 2003 commentWrite an essay on propositional logic. You should include all relevant definitions, and should cover the Completeness Theorem, as well as the Compactness Theorem and the Decidability Theorem.
[You may assume that the set of primitive propositions is countable. You do not need to give proofs of simple examples of syntactic implication, such as the fact that is a theorem or that and syntactically imply .]
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A1.12 B1.15
Part II, 2003 comment(i) A public health official is seeking a rational policy of vaccination against a relatively mild ailment which causes absence from work. Surveys suggest that of the population are already immune, but accurate tests to detect vulnerability in any individual are too costly for mass screening. A simple skin test has been developed, but is not completely reliable. A person who is immune to the ailment will have a negligible reaction to the skin test with probability , a moderate reaction with probability and a strong reaction with probability 0.1. For a person who is vulnerable to the ailment the corresponding probabilities are and . It is estimated that the money-equivalent of workhours lost from failing to vaccinate a vulnerable person is 20 , that the unnecessary cost of vaccinating an immune person is 8 , and that there is no cost associated with vaccinating a vulnerable person or failing to vaccinate an immune person. On the basis of the skin test, it must be decided whether to vaccinate or not. What is the Bayes decision rule that the health official should adopt?
(ii) A collection of students each sit exams. The ability of the th student is represented by and the performance of the th student on the th exam is measured by . Assume that, given , an appropriate model is that the variables are independent, and
for a known positive constant . It is reasonable to assume, a priori, that the are independent with
where and are population parameters, known from experience with previous cohorts of students.
Compute the posterior distribution of given the observed exam marks vector
Suppose now that is also unknown, but assumed to have a distribution, for known . Compute the posterior distribution of given and Find, up to a normalisation constant, the form of the marginal density of given .
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A2.11 B2.16
Part II, 2003 comment(i) Outline briefly the Bayesian approach to hypothesis testing based on Bayes factors.
(ii) Let be independent random variables, both uniformly distributed on . Find a minimal sufficient statistic for . Let , . Show that is ancilliary and explain why the Conditionality Principle would lead to inference about being drawn from the conditional distribution of given . Find the form of this conditional distribution.
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A3.12 B3.15
Part II, 2003 comment(i) Let be independent, identically distributed random variables, with the exponential density .
Obtain the maximum likelihood estimator of . What is the asymptotic distribution of ?
What is the minimum variance unbiased estimator of Justify your answer carefully.
(ii) Explain briefly what is meant by the profile log-likelihood for a scalar parameter of interest , in the presence of a nuisance parameter . Describe how you would test a null hypothesis of the form using the profile log-likelihood ratio statistic.
In a reliability study, lifetimes are independent and exponentially distributed, with means of the form where are unknown and are known constants. Inference is required for the mean lifetime, , for covariate value .
Find, as explicitly as possible, the profile log-likelihood for , with nuisance parameter .
Show that, under , the profile likelihood ratio statistic has a distribution which does not depend on the value of . How might the parametric bootstrap be used to obtain a test of of exact size ?
[Hint: if is exponentially distributed with mean 1 , then is exponentially distributed with mean .]
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A4.13 B4.15
Part II, 2003 commentWrite an account, with appropriate examples, of inference in multiparameter exponential families. Your account should include a discussion of natural statistics and their properties and of various conditional tests on natural parameters.
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A1.11 B1.16
Part II, 2003 comment(i) In the context of a single-period financial market with traded assets, what is an arbitrage? What is an equivalent martingale measure?
A simple single-period financial market contains two assets, (a bond), and (a share). The period can be good, bad, or indifferent, with probabilities each. At the beginning of the period, time 0 , both assets are worth 1 , i.e.
and at the end of the period, time 1 , the share is worth
where . The bond is always worth 1 at the end of the period. Show that there is no arbitrage in this market if and only if .
(ii) An agent with strictly increasing strictly concave utility has wealth at time 0 , and wishes to invest his wealth in shares and bonds so as to maximise his expected utility of wealth at time 1 . Explain how the solution to his optimisation problem generates an equivalent martingale measure.
Assume now that , and . Characterise all equivalent martingale measures for this problem. Characterise all equivalent martingale measures which arise as solutions of an agent's optimisation problem.
Calculate the largest and smallest possible prices for a European call option with strike 1 and expiry 1, as the pricing measure ranges over all equivalent martingale measures. Calculate the corresponding bounds when the pricing measure is restricted to the set arising from expected-utility-maximising agents' optimisation problems.
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A3.11 B3.16
Part II, 2003 comment(i) What does it mean to say that the process is a Brownian motion? What does it mean to say that the process is a martingale?
Suppose that is a Brownian motion and the process is given in terms of as
for constants . For what values of is the process
a martingale? (Here, is a positive constant.)
(ii) In a standard Black-Scholes model, the price at time of a share is represented as . You hold a perpetual American put option on this share, with strike ; you may exercise at any stopping time , and upon exercise you receive . Let . Suppose you plan to use the exercise policy: 'Exercise as soon as the price falls to or lower.' Calculate what the option would be worth if you were to follow this policy. (Assume that the riskless rate of interest is constant and equal to .) For what choice of is this value maximised?
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A4.12 B4.16
Part II, 2003 commentA single-period market contains risky assets, , initially worth , and at time 1 worth random amounts whose first two moments are given by
An agent with given initial wealth is considering how to invest in the available assets, and has asked for your advice. Develop the theory of the mean-variance efficient frontier far enough to exhibit explicitly the minimum-variance portfolio achieving a required mean return, assuming that is non-singular. How does your analysis change if a riskless asset is added to the market? Under what (sufficient) conditions would an agent maximising expected utility actually choose a portfolio on the mean-variance efficient frontier?
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A2.13 B2.21
Part II, 2003 comment(i) Define the Heisenberg picture of quantum mechanics in relation to the Schrödinger picture. Explain how the two pictures provide equivalent descriptions of observable results.
Derive the equation of motion for an operator in the Heisenberg picture.
(ii) For a particle moving in one dimension, the Hamiltonian is
where and are the position and momentum operators, and the state vector is .
Eigenstates of and satisfy
Use standard methods in the Dirac formalism to show that
Calculate and express in terms of the position space wave function .
Compute the momentum space Hamiltonian for the harmonic oscillator with potential .
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A3.13 B3.21
Part II, 2003 comment(i) What are the commutation relations satisfied by the components of an angular momentum vector ? State the possible eigenvalues of the component when has eigenvalue .
Describe how the Pauli matrices
are used to construct the components of the angular momentum vector for a spin system. Show that they obey the required commutation relations.
Show that and each have eigenvalues . Verify that has eigenvalue
(ii) Let and denote the standard operators and state vectors of angular momentum theory. Assume units where . Consider the operator
Show that
Show that the state vectors are eigenvectors of . Suppose that is measured for a system in the state ; show that the probability that the result is equals
Consider the case . Evaluate the probability that the measurement of will result in .
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A4.15 B4.22
Part II, 2003 commentDiscuss the quantum mechanics of the one-dimensional harmonic oscillator using creation and annihilation operators, showing how the energy levels are calculated.
A quantum mechanical system consists of two interacting harmonic oscillators and has the Hamiltonian
For , what are the degeneracies of the three lowest energy levels? For compute, to lowest non-trivial order in perturbation theory, the energies of the ground state and first excited state.
[Standard results for perturbation theory may be stated without proof.]
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A1.15 B1.24
Part II, 2003 comment(i) The worldline of a massive particle moving in a spacetime with metric obeys the geodesic equation
where is the particle's proper time and are the Christoffel symbols; these are the equations of motion for the Lagrangian
where is the particle's mass, and . Why is the choice of worldline parameter irrelevant? Among all possible worldlines passing through points and , why is the one that extremizes the proper time elapsed between and ?
Explain how the equations of motion for a massive particle may be obtained from the alternative Lagrangian
What can you conclude from the fact that has no explicit dependence on ? How are the equations of motion for a massless particle obtained from ?
(ii) A photon moves in the Schwarzschild metric
Given that the motion is confined to the plane , obtain the radial equation
where and are constants, the physical meaning of which should be stated.
Setting , obtain the equation
Using the approximate solution
obtain the standard formula for the deflection of light passing far from a body of mass with impact parameter . Reinstate factors of and to give your result in physical units.
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A2.15 B2.23
Part II, 2003 comment(i) What is a "stationary" metric? What distinguishes a stationary metric from a "static" metric?
A Killing vector field of a metric satisfies
Show that this is equivalent to
Hence show that a constant vector field with one non-zero component, say, is a Killing vector field if is independent of .
(ii) Given that is a Killing vector field, show that is constant along the geodesic worldline of a massive particle with 4-velocity . Hence find the energy of a particle of unit mass moving in a static spacetime with metric
where and are functions only of the space coordinates . By considering a particle with speed small compared with that of light, and given that , show that to lowest order in the Newtonian approximation, and that is the Newtonian potential.
A metric admits an antisymmetric tensor satisfying
Given a geodesic , let . Show that is parallelly propagated along the geodesic, and that it is orthogonal to the tangent vector of the geodesic. Hence show that the scalar
is constant along the geodesic.
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A4.17 B4.25
Part II, 2003 commentWhat are "inertial coordinates" and what is their physical significance? [A proof of the existence of inertial coordinates is not required.] Let be the origin of inertial coordinates and let be the curvature tensor at (with all indices lowered). Show that can be expressed entirely in terms of second partial derivatives of the metric , evaluated at . Use this expression to deduce that (a) (b) (c) .
Starting from the expression for in terms of the Christoffel symbols, show (again by using inertial coordinates) that
Obtain the contracted Bianchi identities and explain why the Einstein equations take the form
where is the energy-momentum tensor of the matter and is an arbitrary constant.
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A1.20 B1.20
Part II, 2003 comment(i) The linear algebraic equations , where is symmetric and positive-definite, are solved with the Gauss-Seidel method. Prove that the iteration always converges.
(ii) The Poisson equation is given in the bounded, simply connected domain , with zero Dirichlet boundary conditions on . It is approximated by the fivepoint formula
where , and is in the interior of .
Assume for the sake of simplicity that the intersection of with the grid consists only of grid points, so that no special arrangements are required near the boundary. Prove that the method can be written in a vector notation, with a negative-definite matrix .
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A2.19 B2.19
Part II, 2003 comment(i) Explain briefly what is meant by the convergence of a numerical method for ordinary differential equations.
(ii) Suppose the sufficiently-smooth function obeys the Lipschitz condition: there exists such that
Prove from first principles, without using the Dahlquist equivalence theorem, that the trapezoidal rule
for the solution of the ordinary differential equation
converges.
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A3.19 B3.20
Part II, 2003 comment(i) The diffusion equation
with the initial condition and zero boundary conditions at and , is solved by the finite-difference method
where and .
Assuming sufficient smoothness of the function , and that remains constant as and become small, prove that the exact solution satisfies the numerical scheme with error .
(ii) For the problem defined in Part (i), assume that there exist such that . Prove that the method is stable for .
[Hint: You may use without proof the Gerschgorin theorem: All the eigenvalues of the matrix are contained in , where
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A4.23 B4.20
Part II, 2003 commentWrite an essay on the conjugate gradient method. Your essay should include:
(a) a statement of the method and a sketch of its derivation;
(b) discussion, without detailed proofs, but with precise statements of relevant theorems, of the conjugacy of the search directions;
(c) a description of the standard form of the algorithm;
(d) discussion of the connection of the method with Krylov subspaces.
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B1.5
Part II, 2003 commentLet be a graph of order . Prove that if has edges then it contains two triangles with a common edge. Here, is the Turán number.
Suppose instead that has exactly one triangle. Show that has at most edges, and that this number can be attained.
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B2.5
Part II, 2003 commentProve Ramsey's theorem in its usual infinite form, namely, that if is finitely coloured then there is an infinite subset such that is monochromatic.
Now let the graph be coloured with an infinite number of colours in such a way that there is no infinite with monochromatic. By considering a suitable 2-colouring of the set of 4 -sets, show that there is an infinite with the property that any two edges of of the form with have different colours.
By considering two further 2-colourings of , show that there is an infinite such that any two non-incident edges of have different colours.
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B4.1
Part II, 2003 commentWrite an essay on the Kruskal-Katona theorem. As well as stating the theorem and giving a detailed sketch of a proof, you should describe some further results that may be derived from it.
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B1.6
Part II, 2003 commentDefine the inner product of two class functions from the finite group into the complex numbers. Prove that characters of the irreducible representations of form an orthonormal basis for the space of class functions.
Consider the representation of the symmetric group by permutation matrices. Show that splits as a direct sum where 1 denotes the trivial representation. Is the -dimensional representation irreducible?
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B2.6
Part II, 2003 commentLet be the space of homogeneous polynomials of degree in two variables and . Define a left action of on the space of polynomials by setting
where and .
Show that
(a) the representations are irreducible,
(b) the representations exhaust the irreducible representations of , and

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B3.5
Part II, 2003 commentIf and are representations of the finite groups and respectively, define the tensor product as a representation of the group and show that its character is given by
Prove that
(a) if and are irreducible, then is an irreducible representation of ;
(b) each irreducible representation of is equivalent to a representation where each is irreducible
Is every representation of the tensor product of a representation of and a representation of ?
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B4.2
Part II, 2003 commentAssume that the group of matrices of determinant 1 with entries from the field has presentation
Show that the subgroup generated by is central and that the quotient group can be identified with the alternating group . Assuming further that has seven conjugacy classes find the character table.
Is it true that every irreducible character is induced up from the character of a 1-dimensional representation of some subgroup?
[Hint: You may find it useful to note that may be regarded as a subgroup of , providing a faithful 2-dimensional representation; the subgroup generated by and is the quaternion group of order 8 , acting irreducibly.]
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B1.7
Part II, 2003 commentWhat does it mean to say that a field is algebraically closed? Show that a field is algebraically closed if and only if, for any finite extension and every homomorphism , there exists a homomorphism whose restriction to is .
Let be a field of characteristic zero, and an algebraic extension such that every nonconstant polynomial over has at least one root in . Prove that is algebraically closed.
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B3.6
Part II, 2003 commentLet be a separable polynomial of degree over a field . Explain what is meant by the Galois group of over . Explain how can be identified with a subgroup of the symmetric group . Show that as a permutation group, is transitive if and only if is irreducible over .
Show that the Galois group of over is , stating clearly any general results you use.
Now let be a finite extension of prime degree . By considering the degrees of the splitting fields of over and , show that also.
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B4.3
Part II, 2003 commentWrite an essay on finite fields and their Galois theory.
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B1.8
Part II, 2003 commentState the Implicit Function Theorem and outline how it produces submanifolds of Euclidean spaces.
Show that the unitary group is a smooth manifold and find its dimension.
Identify the tangent space to at the identity matrix as a subspace of the space of complex matrices.
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B2.7
Part II, 2003 commentLet and be smooth manifolds. If is the projection onto the first factor and is the map in cohomology induced by the pull-back map on differential forms, show that is a direct summand of for each .
Taking to be zero for and , show that for and all
[You might like to use induction in n.]
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B4.4
Part II, 2003 commentDefine the 'pull-back' homomorphism of differential forms determined by the smooth map and state its main properties.
If is a diffeomorphism between open subsets of with coordinates on and on and the -form is equal to on , state and prove the expression for as a multiple of .
Define the integral of an -form over an oriented -manifold and prove that it is well-defined.
Show that the inclusion map , of an oriented -submanifold (without boundary) into , determines an element of . If and , for and fixed in , what is the relation between and , where is the fundamental cohomology class of and is the projection onto the first factor?
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B2.8
Part II, 2003 commentDefine the fundamental group of a topological space and explain briefly why a continuous map gives rise to a homomorphism between fundamental groups.
Let be a subspace of the Euclidean space which contains all of the points with , and which does not contain any of the points . Show that has an infinite fundamental group.
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B3.7
Part II, 2003 commentDefine a covering map. Prove that any covering map induces an injective homomorphisms of fundamental groups.
Show that there is a non-trivial covering map of the real projective plane. Explain how to use this to find the fundamental group of the real projective plane.
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B2.9
Part II, 2003 commentBy Dedekind's theorem, or otherwise, factorise and 7 into prime ideals in the field . Show that the ideal equations
hold in , where . Hence, prove that the ideal class group of is cyclic of order
[It may be assumed that the Minkowski constant for is .]
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B4.5
Part II, 2003 commentState the Mayer-Vietoris theorem. You should give the definition of all the homomorphisms involved.
Compute the homology groups of the union of the 2 -sphere with the line segment from the North pole to the South pole.
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B1.9
Part II, 2003 commentLet , where , and let be the ring of algebraic integers of . Show that the field polynomial of , with and rational, is .
Let . By verifying that and determining the field polynomial, or otherwise, show that is in .
By computing the traces of , show that the elements of have the form
where are integers. By further computing the norm of , show that can be expressed as with integers. Deduce that form an integral basis for .
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B4.6
Part II, 2003 commentWrite an essay on the Dirichlet unit theorem with particular reference to quadratic fields.
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B1.10
Part II, 2003 commentLet be a Hilbert space and let .
(a) Define what it means for to be (i) invertible, and (ii) bounded below. Prove that is invertible if and only if both and are bounded below.
(b) Define what it means for to be normal. Prove that is normal if and only if for all . Deduce that, if is normal, then every point of Sp is an approximate eigenvalue of .
(c) Let be a self-adjoint operator, and let be a sequence in such that for all and as . Show, by direct calculation, that
and deduce that at least one of is an approximate eigenvalue of .
(d) Deduce that, with as in (c),
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B3.8
Part II, 2003 commentLet be the space of all functions on the real line of the form , where is a polynomial with complex coefficients. Make into an inner-product space, in the usual way, by defining the inner product to be
You should assume, without proof, that this equation does define an inner product on . Define the norm by for . Now define a sequence of functions on by
Prove that is an orthogonal sequence in and that it spans .
For every define the Fourier transform of by
Show that
(a) for ;
(b) for all and ,
(c) for all .
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B4.7
Part II, 2003 commentLet be a Hilbert space and let .
(a) Show that if then is invertible.
(b) Prove that if is invertible and if satisfies , then is invertible.
(c) Define what it means for to be compact. Prove that the set of compact operators on is a closed subset of .
(d) Prove that is compact if and only if there is a sequence in , where each operator has finite rank, such that as .
(e) Suppose that , where is invertible and is compact. Prove that then, also, , where is invertible and has finite rank.
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B1.11
Part II, 2003 commentProve that a holomorphic map from to itself is either constant or a rational function. Prove that a holomorphic map of degree 1 from to itself is a Möbius transformation.
Show that, for every finite set of distinct points in and any values , there is a holomorphic function with for .
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B3.9
Part II, 2003 commentLet be the lattice for two non-zero complex numbers whose ratio is not real. Recall that the Weierstrass function is given by the series
the function is the (unique) odd anti-derivative of ; and is defined by the conditions
(a) By writing a differential equation for , or otherwise, show that is an odd function.
(b) Show that for some constants . Use (a) to express in terms of . [Do not attempt to express in terms of .]
(c) Show that the function is periodic with respect to the lattice and deduce that .
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B4.8
Part II, 2003 comment(a) Define the degree of a meromorphic function on the Riemann sphere . State the Riemann-Hurwitz theorem.
Let and be two rational functions on the sphere . Show that
Deduce that
(b) Describe the topological type of the Riemann surface defined by the equation in . [You should analyse carefully the behaviour as and approach .]
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B2.10
Part II, 2003 comment(a) For which polynomials of degree does the equation define a smooth affine curve?
(b) Now let be the completion of the curve defined in (a) to a projective curve. For which polynomials of degree is a smooth projective curve?
(c) Suppose that , defined in (b), is a smooth projective curve. Consider a map , given by . Find the degree and the ramification points of .
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B3.10
Part II, 2003 comment(a) Let be an affine algebraic variety. Define the tangent space for . Show that the set
is closed, for every .
(b) Let be an irreducible projective curve, , and a rational map. Show, carefully quoting any theorems that you use, that if is smooth at then extends to a regular map at .
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B4.9
Part II, 2003 commentLet be a smooth curve of genus 0 over an algebraically closed field . Show that
Now let be a plane projective curve defined by an irreducible homogeneous cubic polynomial.
(a) Show that if is smooth then is not isomorphic to . Standard results on the canonical class may be assumed without proof, provided these are clearly stated.
(b) Show that if has a singularity then there exists a non-constant morphism from to .
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B1.13
Part II, 2003 commentState and prove the first Borel-Cantelli Lemma.
Suppose that is a sequence of events in a common probability space such that whenever and that .
Let be the indicator function of and let
Use Chebyshev's inequality to show that
Deduce, using the first Borel-Cantelli Lemma, that infinitely often .
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B2.12
Part II, 2003 commentLet be a Hilbert space and let be a closed subspace of . Let . Show that there is a unique decomposition such that and .
Now suppose is a probability space and let . Suppose is a sub- -algebra of . Define using a decomposition of the above type. Show that for each set .
Let be two sub- -algebras of . Show that (a) ; (b) .
No general theorems about projections on Hilbert spaces may be quoted without proof.
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B3.12
Part II, 2003 commentExplain what is meant by the characteristic function of a real-valued random variable and prove that is also a characteristic function of some random variable.
Let us say that a characteristic function is infinitely divisible when, for each , we can write for some characteristic function . Prove that, in this case, the limit
exists for all real and is continuous at .
Using Lévy's continuity theorem for characteristic functions, which you should state carefully, deduce that is a characteristic function. Hence show that, if is infinitely divisible, then cannot vanish for any real .
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B4.11
Part II, 2003 commentLet be integrable with respect to Lebesgue measure on . Prove that, if
for every sub-interval of , then almost everywhere on .
Now define
Prove that is continuous on . Show that, if is zero on , then is zero almost everywhere on .
Suppose now that is bounded and Lebesgue integrable on . By applying the Dominated Convergence Theorem to
or otherwise, show that, if is differentiable on , then almost everywhere on .
The functions have the properties:
(a) converges pointwise to a differentiable function on ,
(b) each has a continuous derivative with on ,
(c) converges pointwise to some function on .
Deduce that
almost everywhere on .
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B2.13
Part II, 2003 commentLet be the sum of independent exponential random variables of rate . Compute the moment generating function of .
Consider, for each fixed and for , an queue with arrival rate and with service times distributed as . Assume that the queue is empty at time 0 and write for the earliest time at which a customer departs leaving the queue empty. Show that, as converges in distribution to a random variable whose moment generating function satisfies
Hence obtain the mean value of .
For what service-time distribution would the empty-to-empty time correspond exactly to ?
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B3.13
Part II, 2003 commentState the product theorem for Poisson random measures.
Consider a system of queues, each with infinitely many servers, in which, for , customers leaving the th queue immediately arrive at the th queue. Arrivals to the first queue form a Poisson process of rate . Service times at the th queue are all independent with distribution , and independent of service times at other queues, for all . Assume that initially the system is empty and write for the number of customers at queue at time . Show that are independent Poisson random variables.
In the case show that
where is a Poisson process of rate .
Suppose now that arrivals to the first queue stop at time . Determine the mean number of customers at the th queue at each time .
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B4.12
Part II, 2003 commentExplain what is meant by a renewal process and by a renewal-reward process.
State and prove the law of large numbers for renewal-reward processes.
A component used in a manufacturing process has a maximum lifetime of 2 years and is equally likely to fail at any time during that period. If the component fails whilst in use, it is replaced immediately by a similar component, at a cost of . The factory owner may alternatively replace the component before failure, at a time of his choosing, at a cost of . What should the factory owner do?
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B1.14
Part II, 2003 commentA binary Huffman code is used for encoding symbols occurring with probabilities where . Let be the length of a shortest codeword and of a longest codeword. Determine the maximal and minimal values of and , and find binary trees for which they are attained.
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B2.14
Part II, 2003 commentLet be a binary linear code of length , rank and distance . Let be a codeword with exactly non-zero digits.
(a) Prove that (the Singleton bound).
(b) Prove that truncating on the non-zero digits of produces a code of length , rank and distance for some . Here is the integer satisfying .
[Hint: Assume the opposite. Then, given and its truncation , consider the coordinates where and have 1 in common (i.e. ) and where they differ e.g. and .]
(c) Deduce that (an improved Singleton bound).
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B4.13
Part II, 2003 commentState and prove the Fano and generalized Fano inequalities.
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B2.15
Part II, 2003 commentThe owner of a put option may exercise it on any one of the days , or not at all. If he exercises it on day , when the share price is , his profit will be . Suppose the share price obeys , where are i.i.d. random variables for which . Let be the maximal expected profit the owner can obtain when there are further days to go and the share price is . Show that
(a) is non-decreasing in ,
(b) is non-decreasing in , and
(c) is continuous in .
Deduce that there exists a non-decreasing sequence, , such that expected profit is maximized by exercising the option the first day that .
Now suppose that the option never expires, so effectively . Show by examples that there may or may not exist an optimal policy of the form 'exercise the option the first day that .
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B3.14
Part II, 2003 commentState Pontryagin's Maximum Principle (PMP).
In a given lake the tonnage of fish, , obeys
where is the rate at which fish are extracted. It is desired to maximize
choosing under the constraints , and if . Assume the PMP with an appropriate Hamiltonian . Now define and . Show that there exists such that on the optimal trajectory maximizes
and
Suppose that and that under an optimal policy it is not optimal to extract all the fish. Argue that is impossible and describe qualitatively what must happen under the optimal policy.
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B4.14
Part II, 2003 commentThe scalars , are related by the equations
where is a sequence of uncorrelated random variables with means of 0 and variances of 1. Given that is an unbiased estimate of of variance 1 , the control variable is to be chosen at time on the basis of the information , where and . Let be the Kalman filter estimates of computed from
by appropriate choices of . Show that the variance of is .
Define and
Show that , where and
How would the expression for differ if had a variance different from
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B1.17
Part II, 2003 commentConsider the one-dimensional map , where with a real parameter. Find the range of values of for which the open interval is mapped into itself and contains at least one fixed point. Describe the bifurcation at and find the parameter value for which there is a period-doubling bifurcation. Determine whether the fixed point is an attractor at this bifurcation point.
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B3.17
Part II, 2003 commentLet be a continuous one-dimensional map of the interval . Explain what is meant by saying (a) that the map is topologically transitive, and (b) that the map has a horseshoe.
Consider the tent map defined on the interval by
for . Show that if then this map is topologically transitive, and also that has a horseshoe.
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B4.17
Part II, 2003 commentLet be an orientation-preserving invertible map of the circle onto itself, with a lift . Define the rotation numbers and .
Suppose that , where and are coprime integers. Prove that the map has periodic points of least period , and no periodic points with any least period not equal to .
Now suppose that is irrational. Explain the distinction between wandering and non-wandering points under . Let be the set of limit points of the sequence . Prove
(a) that the set is independent of and is the smallest closed, non-empty, -invariant subset of ;
(b) that is the set of non-wandering points of ;
(c) that is either the whole of or a Cantor set in .
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B1.18
Part II, 2003 comment(a) Define characteristic hypersurfaces and state a local existence and uniqueness theorem for a quasilinear partial differential equation with data on a non-characteristic hypersurface.
(b) Consider the initial value problem
for a function with initial data given for . Obtain a formula for the solution by the method of characteristics and deduce that a solution exists for all .
Derive the following (well-posedness) property for solutions and corresponding to data and respectively:
(c) Consider the initial value problem
for a function with initial data given for . Obtain a formula for the solution by the method of characteristics and hence show that if for all , then the solution exists for all . Show also that if there exists with , then the solution does not exist for all .
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B2.17
Part II, 2003 comment(a) If is a radial function on (i.e. with for ), and , then show that is harmonic on if and only if
for .
(b) State the mean value theorem for harmonic functions and prove it for .
(c) Generalise the statement and the proof of the mean value theorem to the case of a subharmonic function, i.e. a function such that .
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B3.18
Part II, 2003 commentConsider the initial value problem
to be solved for , subject to the initial conditions
for in the Schwarz space . Use the Fourier transform in to obtain a representation for the solution in the form
where should be determined explicitly. Explain carefully why your formula gives a smooth solution to (1) and why it satisfies the initial conditions (2), referring to the required properties of the Fourier transform as necessary.
Next consider the case . Find a tempered distribution (depending on ) such that (3) can be written
and (using the definition of Fourier transform of tempered distributions) show that the formula reduces to
State and prove the Duhamel principle relating to the solution of the -dimensional inhomogeneous wave equation
to be solved for , subject to the initial conditions
for a function. State clearly assumptions used on the solvability of the homogeneous problem.
[Hint: it may be useful to consider the Fourier transform of the tempered distribution defined by the function .]
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B4.18
Part II, 2003 commentDiscuss the basic properties of the Fourier transform and how it is used in the study of partial differential equations.
The essay should include: definition and basic properties, inversion theorem, applications to establishing well-posedness of evolution partial differential equations with constant coefficients.
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B1.19
Part II, 2003 commentBy considering the integral
where is a large circle centred on the origin, show that
where
By using , deduce that .
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B2.18
Part II, 2003 commentLet be the Laplace transform of , where satisfies
and
Show that
and hence deduce that
Use the inversion formula for Laplace transforms to find for and deduce that a solution of the above boundary value problem exists only if . Hence find for .
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B3.19
Part II, 2003 commentLet
where is a path beginning at and ending at (on the real axis). Identify the saddle points and sketch the paths of constant phase through these points.
Hence show that as .
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B4.19
Part II, 2003 commentBy setting , where and are to be suitably chosen, explain how to find integral representations of the solutions of the equation
where is a non-zero real constant and is complex. Discuss in the particular case that is restricted to be real and positive and distinguish the different cases that arise according to the of .
Show that in this particular case, by choosing as a closed contour around the origin, it is possible to express a solution in the form
where is a constant.
Show also that for there are solutions that satisfy
where is a constant.
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B1.21
Part II, 2003 commentA particle of charge and mass moves non-relativistically with 4 -velocity along a trajectory . Its electromagnetic field is determined by the Liénard-Wiechert potential
where and denotes the spatial part of the 4 -vector .
Derive a formula for the Poynting vector at very large distances from the particle. Hence deduce Larmor's formula for the rate of loss of energy due to electromagnetic radiation by the particle.
A particle moves in the plane in a constant magnetic field . Initially it has kinetic energy ; derive a formula for the kinetic energy of this particle as a function of time.
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B2.20
Part II, 2003 commentA plane electromagnetic wave of frequency and wavevector has an electromagnetic potential given by
where is the amplitude of the wave and is the polarization vector. Explain carefully why there are two independent polarization states for such a wave, and why .
A wave travels in the positive -direction with polarization vector . It is incident at on a plane surface which conducts perfectly in the -direction, but not at all in the -direction. Find an expression for the electromagnetic potential of the radiation that is reflected from this surface.
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B4.21
Part II, 2003 commentDescribe the physical meaning of the various components of the stress-energy tensor of the electromagnetic field.
Suppose that one is given an electric field and a magnetic field . Show that the angular momentum about the origin of these fields is
where the integral is taken over all space.
A point electric charge is at the origin, and has electric field
A point magnetic monopole of strength is at and has magnetic field
Find the component, along the axis between the electric charge and the magnetic monopole, of the angular momentum of the electromagnetic field about the origin.
[Hint: You may find it helpful to express both and as gradients of scalar potentials.]
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B1.22
Part II, 2003 commentA gas in equilibrium at temperature and pressure has quantum stationary states with energies in volume . What does it mean to say that a change in volume from to is reversible?
Write down an expression for the probability that the gas is in state . How is the entropy defined in terms of these probabilities? Write down an expression for the energy of the gas, and establish the relation
for reversible changes.
By considering the quantity , derive the Maxwell relation
A gas obeys the equation of state
where is a constant and is a function of only. The gas is expanded isothermally, at temperature , from volume to volume . Find the work done on the gas. Show that the heat absorbed by the gas is given by
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B3.22
Part II, 2003 commentA diatomic molecule, free to move in two space dimensions, has classical Hamiltonian
where is the particle's momentum and is its angular momentum. Write down the classical partition function for an ideal gas of such molecules in thermal equilibrium at temperature . Show that it can be written in the form
where and are the one-molecule partition functions associated with the translational and rotational degrees of freedom, respectively. Compute and and hence show that the energy of the gas is given by
where is Boltzmann's constant. How does this result illustrate the principle of equipartition of energy?
In an improved model of the two-dimensional gas of diatomic molecules, the angular momentum is quantized in integer multiples of :
Write down an expression for in this case. Given that , obtain an expression for the energy in the form
where and are constants that should be computed. How is this result compatible with the principle of equipartition of energy? Find , the specific heat at constant volume, for .
Why can the sum over in be approximated by an integral when ? Deduce that in this limit.
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B4.23
Part II, 2003 commentA gas of non-interacting identical bosons in volume , with one-particle energy levels , is in equilibrium at temperature and chemical potential . Let be the number of particles in the th one-particle state. Write down an expression for the grand partition function . Write down an expression for the probability of finding a given set of occupation numbers of the one-particle states. Hence determine the mean occupation numbers (the Bose-Einstein distribution). Write down expressions, in terms of the mean occupation numbers, for the total energy and total number of particles .
Write down an expression for the grand potential in terms of . Given that
show that can be written in the form
for some function , which you should determine. Hence show that for any change of the gas that leaves the mean occupation numbers unchanged. Consider a (quasi-static) change of with this property. Using the formula
and given that for each , show that
What is the value of for photons?
Let , so that is a function only of and . Why does the energy density depend only on Using the Maxwell relation
and the first law of thermodynamics for reversible changes, show that
and hence that
for some power that you should determine. Show further that
Hence verify, given , that is left unchanged by a change of at constant .
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B1.23
Part II, 2003 commentDefine the differential cross section . Show how it may be related to a scattering amplitude defined in terms of the behaviour of a wave function satisfying suitable boundary conditions as .
For a particle scattering off a potential show how , where is the scattering angle, may be expanded, for energy , as
and find in terms of the phase shift . Obtain the optical theorem relating and .
Suppose that
Why for may be dominant, and what is the expected behaviour of for ?
[For large
Legendre polynomials satisfy
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B2.22
Part II, 2003 commentThe Hamiltonian for a single electron atom has energy eigenstates with energy eigenvalues . There is an interaction with an electromagnetic wave of the form
where is the polarisation vector. At the atom is in the state . Find a formula for the probability amplitude, to first order in , to find the atom in the state at time . If the atom has a size and what are the selection rules which are relevant? For large, under what circumstances will the transition rate be approximately constant?
[You may use the result
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B3.23
Part II, 2003 commentConsider the two Hamiltonians
where are three linearly independent vectors. For each of the Hamiltonians and , what are the symmetries of and what unitary operators are there such that ?
For derive Bloch's theorem. Suppose that has energy eigenfunction with energy where for large . Assume that for each . In a suitable approximation derive the energy eigenvalues for when . Verify that the energy eigenfunctions and energy eigenvalues satisfy Bloch's theorem. What happens if ?
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B4.24
Part II, 2003 commentAtoms of mass in an infinite one-dimensional periodic array, with interatomic spacing , have perturbed positions , for integer . The potential between neighbouring atoms is
for positive constant . Write down the Lagrangian for the variables . Find the frequency of a normal mode of wavenumber . Define the Brillouin zone and explain why may be restricted to lie within it.
Assume now that the array is periodically-identified, so that there are effectively only atoms in the array and the atomic displacements satisfy the periodic boundary conditions . Determine the allowed values of within the Brillouin zone. Show, for allowed wavenumbers and , that
By writing as
where the sum is over allowed values of , find the Lagrangian for the variables , and hence the Hamiltonian as a function of and the conjugate momenta . Show that the Hamiltonian operator of the quantum theory can be written in the form
where is a constant and are harmonic oscillator annihilation and creation operators. What is the physical interpretation of and ? How does this show that phonons have quantized energies?
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B1.25
Part II, 2003 commentConsider a two-dimensional horizontal vortex sheet of strength at height above a horizontal rigid boundary at , so that the inviscid fluid velocity is
Examine the temporal linear instabililty of the sheet and determine the relevant dispersion relationship.
For what wavelengths is the sheet unstable?
Evaluate the temporal growth rate and the wave propagation speed in the limit of both short and long waves. Comment briefly on the significance of your results.
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B2 24
Part II, 2003 commentA plate is drawn vertically out of a bath and the resultant liquid drains off the plate as a thin film. Using lubrication theory, show that the governing equation for the thickness of the film, is
where is time and is the distance down the plate measured from the top.
Verify that
satisfies and identify the function . Using this relationship or otherwise, determine an explicit expression for the thickness of the film assuming that it was initially of uniform thickness .
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B3.24
Part II, 2003 commentA steady two-dimensional jet is generated in an infinite, incompressible fluid of density and kinematic viscosity by a point source of momentum with momentum flux in the direction per unit length located at the origin.
Using boundary layer theory, analyse the motion in the jet and show that the -component of the velocity is given by
where
Show that satisfies the differential equation
Write down the appropriate boundary conditions for this equation. [You need not solve the equation.]
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B4.26
Part II, 2003 commentShow that the complex potential in the complex plane,
describes irrotational, inviscid flow past the rigid cylinder , placed in a uniform flow with circulation .
Show that the transformation
maps the circle in the plane onto the flat plate airfoil in the plane . Hence, write down an expression for the complex potential, , of uniform flow past the flat plate, with circulation . Indicate very briefly how the value of might be chosen to yield a physical solution.
Calculate the turning moment, , exerted on the flat plate by the flow.
(You are given that
where is the fluid density and the integral is to be completed around a contour enclosing the circle ).
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B1.26
Part II, 2003 commentConsider the equation
with and real constants. Find the dispersion relation for waves of frequency and wavenumber . Find the phase velocity and the group velocity , and sketch the graphs of these functions.
By multiplying by , obtain an energy equation in the form
where represents the energy density and the energy flux.
Now let , where is a real constant. Evaluate the average values of and over a period of the wave to show that
Comment on the physical meaning of this result.
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B2.25
Part II, 2003 commentStarting from the equations for the one-dimensional unsteady flow of a perfect gas of uniform entropy, derive the Riemann invariants
on characteristics
A piston moves smoothly down a long tube, with position . Gas occupies the tube ahead of the piston, . Initially the gas and the piston are at rest, and the speed of sound in the gas is . For , show that the characteristics are straight lines, provided that a shock-wave has not formed. Hence find a parametric representation of the solution for the velocity of the gas.
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B3.25
Part II, 2003 commentDerive the wave equation governing the velocity potential for linearised sound in a perfect gas. How is the pressure disturbance related to the velocity potential? Write down the spherically symmetric solution to the wave equation with time dependence , which is regular at the origin.
A high pressure gas is contained, at density , within a thin metal spherical shell which makes small amplitude spherically symmetric vibrations. Ignore the low pressure gas outside. Let the metal shell have radius , mass per unit surface area, and elastic stiffness which tries to restore the radius to its equilibrium value with a force per unit surface area. Show that the frequency of these vibrations is given by
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B4.27
Part II, 2003 commentShow that the equations governing isotropic linear elasticity have plane-wave solutions, identifying them as or waves.
A semi-infinite elastic medium in (where is the vertical coordinate) with density and Lamé moduli and is overlaid by a layer of thickness (in ) of a second elastic medium with density and Lamé moduli and . The top surface at is free, i.e. the surface tractions vanish there. The speed of S-waves is lower in the layer, i.e. . For a time-harmonic SH-wave with horizontal wavenumber and frequency , which oscillates in the slow top layer and decays exponentially into the fast semi-infinite medium, derive the dispersion relation for the apparent wave speed ,
Show graphically that there is always one root, and at least one higher mode if .